Embed Size (px)
Citation preview
!!"#$%$"!&'!())!%$
% *+, *--.
!
" #
$% $%&
$
'( )%
" *+& ,
" *+&
" !( "
" *+& , "
- . ' , -
/ 0 1
21
3 % &
3 %
" 2 3 /
! %
21
! 3 %
! 3 % &
2
4 +( 3 %
% &
$ +(
" 4 "
" 21 "
" 5( -
" /6 % -
" / 7
"" !Æ 7
"- $' 8
"7 , *+ +% 49% 8
- 5( :
- . % :
- . :
7 5(
8 .
8 !% +(
8 *+&
: ; 3 %
: 4+ < % "
: . < "
: . ' "
: . "
:" . -
:- . 0 -
= -
= > -
= 4 7
= 4 7
= 8
21
;
$
4 %
" 2
5(
49% "
+( -
21 -
; +( & & % 7
$ /
/
$ % /
$ /
$ Æ /
" # /
4+ ' % /"
$ % 1 /7
$ 3 /7
$ 3 /8
$ 3 % /:
$ /=
0 3+ /=
. 3 /
. 3 /
. /
" . /
- $ % /"
" # /"
" /-
" # 3 /-
" . /7
- ) /8
7 $ 3 /:
8 $ 3 /=
!
" /
" >+& /
" * /
" # 3 ( /
" !+' 3 /
"" ? 3+ /
" * ' ++& /
"
" . + /
" * 3 3 /
" * 3 3 /-
" * 3 % 6 /-
"" * 3 % 6 /7
"- * 3 /8
" * ++& /:
" * @4 /:
" * 3 6 3 /:
" * 3 6 3 /
" * 3 % 6 /
"" * 3 % 6 /
"- * 3 /
" * /
" ! ' /
" ! ' % /
" ! ' /
"" * % ' /
"- * 3( /"
"- * /"
"- * )% /-
"- * !/ 4 /7
"7 * 3 /7
"7 * A & /7
"7 * ; /8
"8 * 3+ /:
"8 * % +; /:
"8 * ; /=
-
": % /=
": . 3 /
": . /
": /
" # $% &' ( &$
- /
- ! 3 /
- $ /
- /
- . B /"
)
* *
* *
* & *
* / % +; C ==3B *"
*" / % +; C =="3B *-
*- / % +; C ==3B *8
*7 * . ; *=
*8 * Æ ; ' % *
*: ( !/ 4 *
*= * )%;% *
*
8 /
8 / /
7
8 $ /"
8 * 3++& /-
8" ;! /8
+ ,
- . .
8
' & 1 6 D 1 0
' % 0 '6 3 6 "====E 7==== 16 ( FEG H ="
+( % 0 B & 3 +& ' 3 3 & 1 ( < B < & '6 ( <6 6 % & 6 (FI IG H
! 3 6 % & ' < % & & '& ! % %
// 0
; B <E A6 % 6 <EA 6 6 & 3 H
D < 3 J 0 J H
D 1 < B & < K 0 K H
( % 0 KL H J KL H K
. < < & % =0 J
L
L
L
// 1 2
+ 0 1 H
H L
//
3 0 H H J
'6 - H J 0 6 6 6 6 6
// $
D 3 0 $ H
H
H
. ' A 0
0
H
H
H
H
L
H
// 3
; 1 3' D + 3 % B 2 6 %' 0
H
H
H
* B '6 ++ 3% < 6 < & 6 9 < & . 0
H
@ % % <
D 4 3 '
D 4! & ( < 3% A 3
$'0 ; 3' I ' I 3% % ' 3%& I =I
% -6"6 "6-6 -6-
! ( +6 3 (
#' & M%& M
%& M%& % %& H
3%& , 3%& ,
%& H , '<
'< L H
6 3 %& 0
'0 H
0 H
! +
+ % B & 3 %& I " ::=I ( < + 3
<(6 3 3%& (C %6 6 6 % < 3 % 36 %
// 1' 54
+( %&6 < = 6 13% 6 + 3'(%0
67
M M M F= G
M H
H
67
M
@ 3 3 (3 % D + &
//
2 3'( )%6 %0
! " H =
! # H
! $
! % H L
! & 3 ( %& ' &
'
! ' (
H ! "
H =
! ) * M H +
M 0 H
#( 0 H = H 2 B6 H H M
!" #
// !
; ' %& , % # < %D H
H
; , <6
H 5 3%& , + (
3%& N ! . 1 .$1 2 6 $ H
16 0 .$1 H
1$. H
// $'
2' %& , H $ H 3< 3 & 3%& , $ H
0
H
H ' % %O %( H +( $ (6 % (6 3 & ( D ( &
// ! .
; %& ( A ' &' % ' $ 3%& 6 ( ( ( N
D ( H
& ' & ' & 23& +& 6 0 H
$ ( +& 6 0
H $ H $
$$1 H
0 2' + % % % == == % "P & < % -P 5 (3 <' < ( + %N
.3%& 6 6 3 & < + % % ! +6 % H
% H 2 6 3%& +
% H
% H
$ % < 6
"
% H
L
H
"
7 =:
$ % & #
! ' ' ( 1
! ( D +5 ( & B N
D & # ( %0
! / ! 6 +' & ( + & 3 . ( ( B ( & ( (
; ( < 6 % 0
H
6! / . 6 % ( ( < . ( B ( & < & QC ++ Q
! / 6 % (3 1' ' .3 + + < ; ( ( 3 ' < & 6 ( B ( ( % 3 ( H
Q <C Q
( ++& 3 ' 3 3 +' ' ( & %& A
-
' ( )
@ % / 3 % $ 3 &
67
@ 4 & 3 M66 6 % <G F
67
, 6 1 , 6 3 0
H & H &
67
@ % & ! !FG 1
2 6 & 6 % 6 (3 1 %& D 0 & H H
67
@ % & % < F G6G G6 FLF6 GLF ( & H H =
67
@ % & ' & H H H =6
( 0 & G FH =
*
// 67
# 8 # 3 % & 3 F= G 10
0 H , (
//
& +
& %
H = L H
& ( H
// 0 4// !
; & % & % % % % ( ( (
H
& (
// 0 4//
; & % ; < & + & ' < ( 3 0
H
H
16 &6 6 &! < %0
H
=
& G F H H
& H H
H =
& G L F H
H H
// 6 /1/
; & ) ' % *+ ; & ) 6 ,- % . 1 . H & L )
,- H / *- H
*- H
- +*++
+
// 67
; & ) ' % 1 B D 8 2 & ) 6 < 1 0
& ) H & G G ) G +G H & ( ) ( +
D 16 H = LL H
// $ 4//
D 2 & ) 3 10
+ H
0 0
% %0
+ 0 + =
0 0 H
D 1 8 &6 H & ( H L ) 6 + H L +
// $ 4// !
D H & H ) H + " ' 1 2
// 6
; & ) ' % # ! % & '$) ' N
D 1 8 $) H + 0
$) H + H
+ 0 +0
$) H + 0
$) H + H $) H +
; ' % 6 0
$) H + H
$) H + H
, "- .
// 9 4// !
; & % & % 0 & H H
16 ' ' &6
3 3&6 %0 , H
D % ( 3 + (3 1< % 6 6 %6 !+( %& & H < 3+ H
. 1
; (
% +( + 1 ' '
// '
; & % D 1 3 ' & 3 3& %0
, H 0 H
8
& < &
! ; ; & % 1 F F D % & % % & ) 3% F G % F G (0
& F F ) H 4 4 F G
) H 4 H & F F H
H
RB & + 6 ' )
4 ( % +(
4 ) H 4 ; 6
1 3% < F F 6 +( % &
4
H 3&
#( 0 .3 +( 3 1 !3
!+ H
3
%
0 . 3 +( % 3 ; & %
H , H ,,L H , L
; & & ' % '
, L , H , L ,
,
. 3 3 Q % D %
// 67
: / D 3 % & 3 5 %5 H 3& H
% &6 0 5 H
& H
: ! / D 3 % & & 3 6 3 5 6 % 5 H 3& H
: / D 3 % & 3 % H 3&3& H
3 . 3
3 % 3 % & 3 & (3 H 5!
"
// ;'
H 3& 3& H 3& 3& H =
% %
H 5 55 L 5
5
5 H L 5
5 H L 5 L 5 5
6 < 3 6 0
. Æ 3 ,- 0 6 H"
" ! Æ
< (6 < & O % 6 & O % +
. Æ 3 , 0 6 H"
" 6 & 2 6
6 L 6 ( 36 Æ 6 3
S(I ! % R < + < & ( R &
#(0 ! 6 3 6 ( 3 '
// < 4
. % 1 0
H ,, H H 4 , H , H , ,
$ % % $ 6 % 3 !3 ( % 3 6 6 ( %
D ( % 3 % < 6 3' 6 % % & ' ( +& & 6 3 %
. % % & % %0
-
; & & ' %
, L, H , L, L 4,,
& 0& ) 4 % & ) 1 0
0& ) H H 3&) 3&3) H 3F& 3&) 3) G
. % B % 3 ' % ; ' % 6 % ( 3 %
6 & % ' D ,L H ,
//
D % 6 % ) 0 ) H !#
!3 ( % 6 3F) G H = 7 F) G H
// $Æ
. ' % B (1 % % !6 & 3 %6 % & ( Æ & !3 ( Æ 6 6 1 0
8& ) H $ $
H "" "
. Æ ( ' % & ) < ) H & L D % 0
& ) 0 8& ) F G ; & ) 6 8& ) H = ( 3 %
& ) H = 0 8& L ) L H *8& )
; ' & ) 8& ) H D % 3 % Æ
0
7
, L, H , L , L 8,,,,
$ 6
, H
, L
%
8,,,,
//"
; T % < F G ( %
.3< T & 0
H
= (
=
. 0
5 H 3& H H
H
F
G
5 H = 5 H
. H H = T % H H &
//) . 4 :=4
40 *& !&
& < * <%
$ 6 *& H &6 :=40 & !
2 B6 3 *& H & 3& H 6 .40 & 3&
! $ % ( ( % T6 ( & 4+6 3 ( 3 ( (3 < '%
8
$ /-
/"/ # 4
# / : = 2 & H = H & H H
@< 2 & H H F G
, 2 & H H
F= G R( 2 & H H H
2 & H H
2 & H H '
9 = H 9 9
@< ! H
%
R ! H $
!+ ! H
1 1
R ! H '
'
'
$' ! H
= =
#+ ! H $
=
. ! H =
! H
# #
; ! H
= C
U ! H :( K L ( K L
( 3
* 0 2 & C !
/"/ #
# -0 6 , . % & < 3 %&
# 0 D % % = $ 3 6 & 6 ( 6 3 ' 3 3 3 %& . ' B &
# '0 . ( 3 <BV %&
# 0 !3 3 % ' < %& ! (
:
# 0 D % & 3 % ( = 9 H D ; +& (3 %& & < WV ( 3%& * ( ;1 3 %& ; 3 H 3 +( 3 3%& * % 3%& D ( 3 %& * 0 H H )
)
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
1 3 5 7 9 11 13 15
f(int (x))
0 2 & 9 H =
. ' & 9 9 &9 L 9
# 0 $ 1 6 &
# 9 R. 0 !3 D & 4% (6 786 % D %0 # + 0" H D % & 3 3 % S%I $ B ( 6 +& @ % % & % 0 ; & % &
' & ' 6 Æ 5
% & &
5
5
# 0 $ 1 6 & % ( .3 % 4*, . * /-0
' < . ' R
H
# >0 ! & 1 $ & ' %0 & U &
=
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-3 -1 1 3
f(x)
0 2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 3 5 7 9
f(x)
0 2 ' & H
: &( ' D ( : & <0 : & ( % C : ( & ( 1 % C : H & ' &
# ?0 ; % T & ) %
) H "
$ ) R & 9 H
.
' $ A6 ' & 3 ' %& 6 < ( 3 )& )L& 3%& . ( & % % ( %+ (
# 0 . & ! 1 %
H
& % % 316 ( %
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-3 -1 1 3
f(x)
0 2 U & : H
. ' % & % 6 % % & L D & R% 9 H $ $
# 30 !3 6 ; H L & %
!3 %
# &0 ; & 0 F= G6 ) & ( ) $
& 6 % H
; & ! ( 3++&
0 /-
$ (6 % & % %0
X& H
&
H
& X&
H
"*
2 6 <(6 & 3 %16 ( & B& 6
X& $
;
1 % 2
/*/ $4 '
D 3 & 3 % (6 & & %& 31 %6 ' ' '
$4 ; % & # 6 & % # D ( & % % % & % %
$4 D ( & % % % & < = ( & & = ( <
! 1 +& , & & $ ( +&6 ( 3% 6 <( % 3%& % % ( 3% 1
$4 D ( & % 3 % % & 3& & = % 31 . % % (( H
. % 3 ( % 3 % ( ( % ( ( % ! & %
0 *+& 2 4%.0 ; & %
% % = D
( % " " '6 %& 0 = = . & % ; & ' % & H . %' =: 3& 23& +&6 % +
& H 0 & H
( (
! < 3' % 8 0
& H
( (
H =" ( ( =7 H =7 ( ( =" H
=-8:" ="-7" H = ; P
/*/ !
! +& < 1 '
!0
; & % B 3 3 %
"
$ % % % =
( . ' +(
% %
6 +(6 3 $
@ % +(
3
! +&6 % % 3 % +& , ! +& 1 3 (3 3' ( 6 +& 3 D & % ( +& % B % A6 ( % + % %&% % !3 < +& & #
!0
; & & & % 6 < B 6 3 %
; H
< %
& ; % 0
%*
H =
&
=
. . ' ( % "*
S< I % . % (3& < 3 % 6 1 6 % 3 6 6 & A <6 % !3 & ( & % & + &
$16 ( +& 3' % D ' % % !+ 6 % ( !+ ( ( 3
3 4 .
*& % 6 & + 1< 3% +
+
/+/ : 8 4
; & + 3 % & < ; ) % 1 0 ) H & ! % < 3% F= G ; < R (0 >+ H F) ( +G H +
; + + + 3 % < F= G .+ % B % ) 6 Æ % ( & % + < 0 & H ) H +
/+/ # 8
. 3 + 1< 3 % (( 3 + 1< 3 % < = <6 % + D +6 + < ( + '
/+/ #
H
= = D %0 H
H
.
+ < % 0 H + ; + + ( ( < (6 0 + H
D H ? +
/+/ #
H
% B D H = & + < +( +6 < 0
+ < H L
+ < H
% 3 % B ! + ( & =
"
/+/ #
H ' '
D < ( % ' %& <
% ' D & % % ' . % & 9 30
( 9 (
% H ? + + 0 % < F=6G % '
/+/" # (
D +& . X) 3 % ) % ( 3+ Æ 6 ( ( % ) D Y 0 % < F=6G 2
3) H 7 ) H
. % 1
%
+ % % 6 0
H L
+
$ (6 ==
5
' 3 %
/-/ @
.3+ & .3 & +( & <( 3 % 3+ 2 3 % &6 3+ & 6 % 6 % &
. 3 + 3' & @ < < 3< &
-
A % ' 3 $ %+6 & +( + & %% ( 6 3+6 1
; '
H F L ? *G
& F G & == 6 < 3+ 7
/-/ :
16 % < 3 %
Z 0
H
; % (6 & 3 +( 3 ( !3 ' % '& ( < & (
; 3 3 + 6 Z H ; 6 % Z H
( . : - 7 = % & = - 7 : ( "
.( 3 (3 6 ++ F G ( ( =" =" D Z <
Z H L ="
/-/ :
16 3 % %
0 & H H % & H
. 3 ( 3 1 ( % & % 6 & H H = !6 % + 3 (3 % % ( 3 %
7
/-/ 1
2 (6 % ( 3 + & 3 ( ' 3 0
. ( 0 & H
. +( 0 + H
& BV % B %
8
6)
D < ! 3 < 46 + & 3 3 '
+6 & 6 + 4 ( % D 0 % '
D < !0 6 %6 !( % < ' + 6 % 04 4 Æ 8,, D 3 3% (3 3 ' < B ; 6 & 6 ' A 6 3 8 3% 0 H
! < 6 < 3%6< %6
// &
@ 6 6 36 3 & ( 3 0
0 H 0 L L
D < % ++& +( 3 ( ( 3 % 3 ; ++& 3 % 6 & 2 6 % % ( & % D (
D ( 6 ( 3 @ H 3& 0 H 0 @ D 1 H @ '
//
@ ' B & 3 % 3 + <E (( 3'
! ++& +( $ &3(% ( 0
34 H 4 H )
)
4
% & % & 9
//
' D %1 < ( <3% @ H @
@ H =6 % & % 0 = H H % 6
8= H 23 6 @ @ = 8@
(6 3 -
( @ 1 6 % % ( D 0 - @ H =
2 ' +6 &
%0 @ == H 8@
; + ( 6 @ 0 @ H@ H 8@ H = 4 B6 3 % 3 Æ 3 <
// : 4
2 3 % % & B 6 3 % 3 % B % @ % & & & & & 0
0 ; H 3F&&) G & ) 3
3 %
40 H 3 F& 3F&G& 3F&GG; % 3+ 3 6 %
% ( A @ H 1[@ & [@
3+ 2 6 3F&G H 3F&G
; H
;= ;[@ ; [@
;[@
;[@; [@ ;[@ ;=
@ ( +( 3 (
*O
H
8 8
8
8
8 8
& 8 H 8 1[@ ! ' 31 & D < (3 % ( 1% 7 H = 7 7 =
@ % ( + 0 3 Æ 8 % & 3
8 0 ; 3 A + % <) H & & (( < 6 %0 3) H 3& H
9 4 0 D < 4+ 6 < ) H
& 3& . % ) % &
6 :0 D 4+ & & ( 0.& H && ! 3 3 % & + % %
// 6
. 3 < < 3 (0
\ H ";@ H
;@
- @
! ! >5
/-
//
@ 3 6 % H 6 % H 6 & !+( &
H
. 0
H
"/
"
; % 6 % %
& 3
$
% .
< 30 H
2 3 % +6 3%0
H
8
8
H
8
. + +
.3 6 3 6 +& 6 ( < & ( 3 +& 6 3 6 &
2 16 (3 & % 6 < 3 3 BV ( ( ! 3% %6 8 H = 6 ( %(
$16 < 1 ' 3 ! %1 % ( % B % & 3 % &
//
. ! & < 3 ' +& 3%&
0 +(6 & 6 3(6 6 2 +& % B ( 3%& ( & 3 ( % B
; 3%& 96 % %& % 1 @ % 9@ ! 3% 3 %& @ 6 3% %& @ % 9@ D ( A ' %& % @ !3 0 @ H "
" & & H 9@ 3%&
% @
; 3 - % 3% ' %&<6 % 6 '
- H 9'06 - =
@ B 3 2 5 0
H
5 Æ
. < 3 3 %% 5 H 0
;@ H 9 L 9Æ@ \ H 9L 9Æ
@ 3 <(( * 3' 3 1 * . + 1 1 & 3 3A '0
+ H # * H
5 *
5 H 6 + & \ H \>& > < *
// :=4
2 6 3% < 3 6 ( 3 . :=4 .( 6 %A :=4
6 ( (( 3 ( ( ( ( '( % 0 9% %10
H
"
'6 49% #& ( 3 < # D %0
3F G H 3FG
H G ! & ( ( &
!6 < $ % <3( !+)%
H
( (
. +& ( B . ( 6 & (( 6 3 ( & 3 ( %
"-
// 67
@ ' +& & % +6 % % <
$'0 ; +(
2 6 % & ( %0 & N
& (6 Æ ; - H F- -G 3 . Æ -
8@ H3- -- 3- 3--
00
! Æ < & @ 6 B 8 ; < & 6 ( (3 3 ( & $ 6 3 2 6 <3 < BV ' ! ( % C ' +& +( ' < ( (
-
// & ' ! ! 4
D - 3 & & 6 3 6 ,6 1 0 H
$'0 ; & B % 0 =P % & = 6 =P & -= 6 =P &:= =P & = . & & 3 &
- H = L = L = L = H = L = L = L =
. = L = L = L = & 3 < < D <0
- H
. Æ & . 1
8 #6(0 D % (
'6 - H = "- L .3 3 , & 0- H
1 6 #6( 3' <
- H L L
L L
L L L L
7
D & ' 30 3 ++ & % 3 % 1 & 3 ( % % 3 & & %
//
3+ <6 % & ( + %
'6 A & % 3% & . % B ( + & ' +& % ; % ( & A %
; % % & + % &
D A6 3 6 < ( ' % 3+< % & A D % (
BA H
16 < 3 %6 % ++ & ( 6 6 #6 6
0 D % +& & < !+( & & ( ' ! %1 +( 6 & & + ( < +; ( < 3++&(3 & . & 6 % 3+6 % < & & % ( % B 1 & (
B H
B H
/
B H B H
5 N $' N
; '6 & < 30 ( 1 +( N
. 6 3 ' 6 &6 % 0
0 D + ( 3 ( & % %
0 ; 3 & 3 +6 + +6 % 3 + & 3 D 3Æ
40 ; 3 % & &6 % 3 B % % &
0 * D 3+ &% ' <
0 2 &6 ' D + (3 & %
! A & < '6 3 % (6 6 % '
// 4
@ % % % % % &6 0
0 4 A ( 4
; 3 '+% 3+ & &6 %% &
//
@ ( +( & %% &
3 H A
@ (
/
3+ 0
3 H A
0
T 0 A 0 X& H
% %
%
T 0 A0 H
% % %
%
T 0 A 0 H
X& % %
% % % ! (
T 0 A 0 H
X& %
% % %
. A ' ' < . '& < (3 < % < 3 %6 3
// Æ
. % 3 B6 % 6 % < D 3 ! 4
.( 3 ' 6 ( Æ (
7 ( 7
@ & + 1 ; & 16 3 3% & & % % A . ( 3 E 3 < % 3 . 1 / .1/ ( $30
7 @ A
'A
& 'A6 ( 3< 3+6 1 0
'A H 3
B??& A
BA
?& A 8 4 0
? A H
A A A
/
% &
@A H 3FG @ A HB@A
BA
; <6 4/,
3'( . 3< = ++ # + & ( +( . + + %0 3 3 ' ( 3 %16 3 ( 6 BV < % & + % ( . &% 3B 3 +& % B < + & +A ( 3 ' 3 + < 3< + B %
++&6 + 1 +( 3< % % + . & ! ( % 3 < & < % 3 & < (3 ! 6 ( & + 3 3<(6 ' ( 3 % % +O ;+6 ' < (
// 3
. S I & < ( R$ ,' :"@ & ++& ( ' SI0 % 6 !3 '& ( (
6 % 3 ' D 1 +( '& ( 1 % 3 & 3+
! + % ( 6 ( < % '& 3 +(6 +( ( B 3+ !6 1 & 3 3 % %6 6
6 + 3 1 3 %
'& $ <6 1 ( % '& % D ( 3
&
% (
/
," &
. & 3Æ D 3 % ; & % A ( 3 & A ( . & & ' ( < % + 6 % % ( % & A
; %6 (
C H
3 % C& H
&
&
&
' & '
.3+ & ++ % A ( ( 3 % 3 . 3 3+ ( B 3 + ( ' & '
C& H C H
A
. + ' % & ++ % A ( ' ! 3% % +6 % A ? A 8 4 . % BA ( ' < % ? 0
B??
BAH = BA 0
B??
BA( =
.3 + < ? ' 3 & 3 6 B < +
8 40
?C AC H
234 525
C H =
3!
2334 525
"H =
3#
34 5
234 525
$H 3
!2334 5
25
"!0 ; ' Æ 6 + ' %
/"
!0 .3 Æ ' 234 525
H AF @AG & A % D (
7 FG H
%%%%%@ AA
%%%%%! + & +( & 3% 3B < <
0 ; & % .3 < % 6 + C H 6 < % +0
?C H
H
?C H D
??C H D
&
'
B??
BH = B 0
B
H =
B H
. +( 3 Æ 3 +( 5 N
B C& H
&
& & %
3FBG H 3F
&G H
3F&G H
3 +( .3
$
/-
)
! % + % < ( 3 D ++ & 3% F G % ( & A % % % 5 1' ! 3 ' ' 3
F ( A ( G H 5
.3% F G 4 7 5 Æ 7@ % 1 3 ( 3% Æ 1
. 6 3+6 & & ! 3' &6 % <( 3 6 3 63 %
//
; % & & % =Q D ++ & % + D ( % & & 5 6 H
' % % 6 % (
1
; H C % % =Q
B &6 !3 6 3& +& 6 %
% 3
!' % ( 3+ Æ = (
! 3% 1 <0
( H 5
& 5 ( 6 % ( 3
%
% % E
H
0 =
D 3% 1 <0
F ( A ( G H
( ( L
H 5
/7
. % H
. % = ' A & ' % 0
0
F ( A ( G H
( ( L
H 5
0 A6 ( ( % 6
F ( A ( G H
( ( L
5
0 ; + H == H =- 5 1 % ( % 1 =: ( =PN
0 =-
=- =
==( ( =- L
=- =
== H ( =-
== :( H =:
6 & ( H =:" H - ".3% & :=P 1 F=": C =-8=8G
$
//
2' & %0
. % ((
. % 3 &= 6 % 3& +&
; % & % 3 D 3 + % ! 6 3% 1 0
F ( ( G H
#
( ( #L
H 5
/8
& # +( & 3+ 6 % '
. %
. % B 3 < 3% 1 % % ) H "
$ . %
. %
2 6 B 3 % ; 3 ( 3 0
H
#
! & 6 ( ( *
$ &
% H
# . % % ) H "
; & .3% 1 0
F ( ( G H
#
( ( #L
H 5
& ;
6 B 3+ ( 3% 16 Æ 1 5 6 ( < 3 P % D $
6
( &
D + # 3
// 4
3 ( 3 % 3 % & & 3 + %
; & 6 3% 1 & 5P 1
C
% H
&
( 3 6
6
&
/:
; . ( *
$1 + &
.3% 1 & 5P 1
C
&
( 3 6
6
&
D % 0
! (
"H
(
H 5
% 6 3& 3 0
H 5
% H
#
+ & 3 1 (3 % & A ' ( % ( & ! &% ( 3 ' . + ( 36 8=" & 0
2 3 4 2
// (
; * 6 +6 ' ( 3 % ' . & 3 3 & ( 3 ; & * @ + + !3 & ++ ( ' F* G ! 3 % % +& ,
/=
F * G H F* G F* G H F* G F G H F* G FG FG
3& 3 F* G H & / / / & /
. ' ' < 3 6 ' FG ; ( ' 6 % 3 ' F*G FG
2 '6 F*G % 3 FG 3 6 ! '& ( < (3 % & ( ( 3 3 6 +
// #
4+6 3 6 3 (6 (3 (3 Æ 3 ( B
3 2 6 ( * &% 3 +& < 6 ( * H L 2 6 3 3 F G ; 3 % +& 6 < ' F G FG . ' & ( ' 3( $ A6 1'6 & 3 < < & ' 3%6 6 % < ( ' 6 6 6 ' & < 4 %3+ & 3 & % N
// #
.3 3 < 3 Q &1 < 6 < ++& & &. ++& ; * 3+ B* &
;0 F* B* =G H F* B* ( =G
2 % 0 F* B*G ( * B*
/
0 F*G H F* B*G
6 % ( 3 D <
% ? H D
& H * B*
D ++ ' % 6 ( & + ( +
2 3 (6 < ( 3 & % % % < *+&
. & +' ( 3 & = < 3 = ( < <( 3 3 & < & 2 6 6 :"P % % & < 3
D B & + '6 !+6 '
// #
3 3 & + ; &; 3 < 3++& 3 6 & 3 6 3 ( ' H D6 & + ' % 6
! ' % B B ' ( % %( + ' % . ' F ( F ! * < < 6 & 6 ( ( ( =" ( 6 '
// #
# D % ' F*G H 2 & 3
1& 2
. ' < 3 3< B & % & & Q
/
H
8
* B*1
% 8 H ! *2 & 3 1& % % +(
; G H 7 . & &
& (0
G
* B*1
BB*1
BH =
! & 3 % 3 < +' 86 ( 3 4
:! #0
!3 & 3++& D 8 H G H
#0
$ & 6 8 H G H * ( & 3
:! $ B #C0
! 3% % 6 & !+6 '
8 H ! L G H
. & & <
% & %
:! @0
8 H
(=" ="
G H
(
2 &6 ( 3% & < (( 3 < 3 < & & . .
:! =0
/
. & *9 B ( > ' 3A+ +'
8 H
&
*
'
G H
&
*
' (
=
. % 2 B > . % 1 % 7:" % ( & . % < ( 3 & 3 D & 3 *9 6 4 < 2% (%
H 8- # #
D % '6 5 P % @ % 5 H "P . '& ( %5 H "=P
3 ]96 :70
. # 3 . < %0
H
;
& ; 3 . < ; (
; H = '6 U' < % ; H ;
. )%;% ; 3 ' #
# ! #/:/&/ . 4 < ;(6 #^6 :8 0
. % & %# ;
6
6 ( & 6 ' % %
F ! *
/
//" 4
* ( % + 6 6 %6 < < Æ < ' 3 6 3
%2 6 ( (1 3( +( &6 3 !+( A 3 6 % % & 3 D 3.3++& (3 < ( < 3% ( & '& +3 . B (3& % 3
' 3 3 +( +(& 3 + H D 3+ %0
& H 0 # H
! 0 H #
! 0 H8
< + % G
% L 0 # H
9
9
" %0 ; % '
= (
2 '6 % B ' < F F $ (6 3 & %' = & +( . ( % % % & & D 6 % 6 (5P ( ( 3 < 3 ++ 6 1' % 5 H "=P
72
. 3 & & < & & !3 3 @ + & & < 3 + ( < ( &+(
/"
// 0
; < ( C H H +
D + ' B % & ( BC H
.3 < 3 % & 3 C + 1 H 6 < + < B 0
H
8 BC +
H
8
D ( B 6 ( < %
// 3
% % (3 & < +' +' ( 2 6 % % ( % & 8 H D &3( %0
B B C
B
H = H
;
BC +
H = H
L L L H
+
! & 6 ( B < & D ( % ' 3 # ( % C D ( & 1 3 . % % & ++ #
. & & ( D < & +(1( ( .@ + ' F + R; ] + % ' F &
/-
3 & % +(6 <O % 3 ( & % &
// #
& & D D H + B H 5L : . & & 30
5 L
: H
+
5 L : H
+
! & (
H =
D ( + 3 % & ( 3
B3F&G % B7 F&G . 3'3 B3F&G B3F&G H B7 F&G H = ( (% & (3 < ( 3+ ( . & <0 (
B3F&G 5L B3F&G : H B3F&) GB3F&G 5 L : H B3F) G
( 0B H
! ! ! ! !
H / #
B H! ! ! !
! !H B3F) G B B3F&G
. % & ) ) H X & L X & X X % %
D % ( 0& ) H 8& ) 7 F&G 7 F) G 7 F) G H X7 F&G D
% B & % % X 0
B H 8& )
)**+7 F) G
7 F&GH 8& ) X
.3 < % & ) < 8& ) H < 6 3 . Æ ( 2 B6 & 6 ( 3F) G H X 3F&G L X 26
B H X B3F&G L X B B3F&G H 8& ) B3F&GXL X
.& 6 3 3 ( + !6 ( B3F&G X % 3 & < .3 X % ( X
/7
$ * 8
2 & 3 ( % 3 6 1' + .3 % & + 2 ' ( 3+ & 6 (3 & + 1 B & & 3% 3+ +( % 6 ++ & & % 3 3 B ( 3 & +( <6 ( ' D 3
+ +( ( ( 6 7 5
; A 3 A 6 2 B6 A 3 B ' & D 3% 3 % + + ( H H . & % % A 6
. & 0
A H A LD+ %A
! 3&N . % ( 3 A . %A L & & 3 & . +%A 3 D ( 34 % ! % % % & & 3 ! & & 3 < % %6 D 5
. ) % 3 ; 3 H % & 6 % % <6 1 3%6 3 & 36 B & < 6 1 6 (3 1 3 ! ( %0
D H % H L %%
.3 % < ( 3 B
(3& & 3 3 ( 3 ) 0
/8
H ' D%
' 6 H H % H D < %0
A H A LD+ A H DA LD+
H: *
: *
D H*
: *
D ( 3 & # & ( 3 3 % A 3+ #
0 .
% % + < ( 3 1 ( % & ' %
3 ( % & & & H H % & H
67( ; & % D & % ( < &
H
%
H %
H
! ( % ( 3% F G % &
! % & 3 +N D + % %6 1' H
D ++ 3%
6 %) 6 % .3 1 < & 16 3%
. ' % ' F! * & < 3 % + 1 6 B < % & ' 3 ' % 3
/:
1 .
% % ( 3( ( % ++& 3 3 (6 3 & % 3+ ; ' & 6 % ( Æ & < 3+
. + 3 % & ' & 3 +
. & + & % & 6 3 & & & % . B (( '
@ & & <B D ,0 ' H F +
L +
F6 % 3
& <B0 B H " 0 + ( L +
D
B H
D
,
& D < 3% F$ $F0 D H = (
6
D H (
D
+ % '
! + & ; 3 % < 6 < '6 < D6 $ (6 % ' ( & <(
D H
( D H
( "
! 3$+9% +( . , ; B 6 &3 3+0 , < 6 B & &6 , 6 B &
, ( 3 + ( , B & 6 % % (
3 ( D % D & <% 3 ' % %6 ' 4.0 D H
/=
9
// @ !
@ ( @ ( + ' ++& % 3 + 2 ( 6 ++& % 3 6 %6 3 ; I I ' ++&6 % . & + I I %0
I % I %I 5 :I 5 :
5 : ! ! !0
5 I ( I %
: I ( I %
! ' 6 5 6 : . < 3 5 < & & ( % & 3 3++&I ( & % ++& (( < & . % 5 : + ( Æ &
D ( :
2 ( (6 & 1' 5 % =="6 == = < < ( & &$ A6 I % B & 3++& I ! ( < ( I B 3++& < ( I % & 3++& '6 % I 0 # H # ( %< & 6 I # H #
! ( < ( 3% 3 3% <' ( % 5 5D & + 5 !6 3 <( ( 3' + & ++ % 5 ( 3++& I %
/
//
% 3( (6 3 1 & $ A6 ++& <6 ( 6
@ 3( ( ++ A ' 6 % A2 6 O < % 3++& < 3 <
@ 3( ( ++ & % < & & % . O 3++& 3 B <
! 3 % ( '
// 3 '
5 + N 5 1'6 < + 4 6% ( 3< & 6 & % +' '++& . % B < ++& % I 1 % & D '6 3 H 6 3 % % ( & I 1 I D H & 5 H I H 5
D 6 3 H ( D % & '& &0 H I H 5 H I H : . O 3 & 3% ( A % + ' ? 3+ . ( O ' % ( A B < + 3+
. 3 ( 3' D + & %0
!+' I I
2 %
( < I
! ( < 5
! % :
! ' %
/
! 0 I
// $
& A % & % % ++& 2 6 ( < 3 : 6 B % 56 1 % : $ A6 A 3 & &
. ( 3 ; 36 % % A 3 3 ( 6 3 (% & %&%
; %1 ++&6 + B
. '&& D6 3% & % $ A6 '6 ( % % ( % (3 ( % & & 2 6 B & 6 ' Æ ( ( I % . % : 5
// D
6 % & ' +6 < +( & ' +; 6 +' + 3 ? +' 6 ' +
; 3 &% 36 'B6 ' + 2 6 +( + '& A <6 B % ( B % & ' A6 ' A '
. +( 3+ 3% 3 ' % < ( 3 $ & 6 &3 '6 %
/
2 ( 6 (( ( ! % '+% #O% & % + ( (
! & ""
// # 9
; & % A & A & ? A 3+
@ ' ++& 0
(I 0 A H AI 0 A H A
; 3 & & 5 D % ( 3 5 & 3 0
H I H 5 H
:? A
D ++ 6 ( '0
H I H : H
:? A
. +&
!0 . ( 1 3 (0
? A
? A 6
$ ( +&6 0
: 5
: %
//
; & % # 3 % 3 + 6 % # & # & #6 ( 0 (
I 0 # H #
I 0 # H #
/
. < % 6 6 3+ 0
? # H
F
G
? # H
F
G
. ( 1 ' < $ +6 <0
#
# ?
$ 0 X H
6 0
&X # L#
'# # ?
; # ( #6 &0
X # L#
?
# #H 9
. ( 1 3 X 9 (3 < 6 3 5 ! 1 0 5 H II I X 96 5 H & 9I & & % X & % 6 & # 3 $
D I %
5 H & 9 % & 0 F# $G
5 H
$
'
$
. ( ) H
$
0
5 H ) '
$
% ) 0 F= G
; % 5 1'6 6 % % '
$ 9
. & 0
/"
& X 9 1 I & I/
(%6 % 3 '& &
: H & ( 9I H
) (
9#
% ) H
$
%
//
. 3( (3& 9
5 H
& #
$ 9#
& 3 3
. ( ) H
3 3 % % &
H
X
6 ) % % ; & ! ( 0
5 H ) '
$
% ) 0 ;
.& 6 B & ; % % 9 . & B
2 B6 6 & & 3 '& & &
: H & 9I H
) (
9#
% ) H
% ; &
// 4 <
; & % # D ( 3 ( ' % % 3 + 6
/-
% % & &
6 ( 0(I 0 H I 0 H
.3 %
H
#
D $ $
. < % 6 6 3+ 0
? H
!
"
? H
!
"
. ( 1 ' < $ +6 <0
?&
'L
#
?6
2 6
&?6 ?
'
. % 6 & 3 & & . ( *
$
& . %
// 4 <
! <( ( * B +& 6 ' %0
. % H
& & ( ( *
$ &
. ( 1
H
H 5
. & 0
& 1 I & I/
/7
//"
; & & 3% & & D ( % ( ' % % 3+ 6 ( & % 6% 1 5
3+6 3 +( <( ( H
& 3% & 3+
. ++& (I 0 H I 0 H
. & ( ! " I
( ! " I
& (
3 % B B +& ; 3+ Æ (6 =6 ( % 3
! ( &
5 H I 0"
% 0 F
G
; 3++& I6
5 H
H
)
& ) H ;
%
. % (
.3 & 0
: H
)
& ) H ;
%
/8
! ""
// E:
2 6 ( < B 3++&0
I 0 A H A
6 3++& I < 3 3++&
I 0 A _ 0 A _
. ' 0
I 0 A A
I 0 A ( A
, '
I 0 A H A
.3 & & 1'6 ( H & +( % A _6 % : 3 & . : H *A % A _ 3Æ
. 8 @< 4 ^< E: ( H % A _
!0 ;3 ' @46 & &
6 I B B 5 A % A _ D % ' 5A 5
. +& .+ 3' @4 %0(I 0 A ( AI 0 A A
(I 0 A A A AI 0 A ( A A
6 3' @4 0 I 0 A A A I 0 A A A ( A6 6 I 0 A H A I 0 A H A
(( ' '+%6O% & +
. & + 3 % % I & I E
// <
/:
; & % # % % 3 + 6 % + ' ++&0
I 0 # H #
I 0 # ( #
! 6 3 & & 5 1' 6 # +( X . & ( % % 3 D &0
5 H
& #
9#
% & 0 F#$G
D ( ( % 9 % # 3++& I 3 ( <
. % ) H
$
A B 3
. % 3 B 3 '& &
; & % # % % 3 + 6 % + ' ++&0
I 0 # H #
I 0 # H #
! 6 3 & & 5 1' 6 # +( X . (( 3++& I ' ++& 0
I 0 # ( #
I 0 # #
+ ' ++& 9 9 D ( @4 ( 9 3
. ( ' ++& I I
D 5 H & 9 & 9 H & 9 L & 9 H 5 L 5
1 % 9 9I !6 & ( 6 5 H 5 H 6
( &
/=
% 9 ( & # !+( < % 5
5
H
) H
& #
9#
% & 0 F# G ) F= G. % 3 B
3 '& &
// <
. ' 6 6 B . % ; &
// 4 <
. ' 6 6 B . % &
// 4 <
. ' 6 6 B . % &
//"
. ' 6 6 B . %
2 6 3 < ( ( '% 6 +(6 3
. ( 0
6
& 6 =
/
0 ; + == % 3 6 =P <% & 33 ! 3++& ( + <% N
! & 3 3++& 0
I 0 H =I 0 H =
% 5 H =="6 H :- 3& (0 = :-
H ==-"
H H F=-" =:"G
! = H ==6 I 5 H =="
!
; & & ' % 1 ' & % . & & 3 & A A D + 3++& I ' & 'I 3++& I ' & AI6
I 0 A H A I 0 A H A
A 6 3 + & & A A D ( % & & '%
$ I %6 ( & & 56 O & ' % ( (
. ( H . H5
& . <
; . & O 6 I 6 I ( 5
// $
; & & ' 6 3 D
I 0 H I 0 H ( 5
/
D ' + ( < # # 3
; 6
- H# #-
$
L$
D I ( 5 - F
G & %
; 6 < +
; ' & =6
- H# #-
L
D I ( 5 - F
G & %
; < & = H
- H# #
B
L
&
B H
L L
D I ( 5 - F
G & %
; & L
; < & = H
- H# #-
L
D I ( 5 - F<
<G & %
<
; & C 3 +
!
L
"
L
/
// $ 4
% B ( 6
I 0 H I 0 H ( 5
D B H 6 B
H
- H
D I ( 5 - F
G & % 6
+; & L
#( 0 H
;
// $
; % 3% 3 & % % D ' + % % D +6 3 (
I 0 H I 0 H ( 5
D ( % '% D
B H
- H
D I ( 5 - F
G & %
! &
! < & & + +
* I 0 A H A I 0 A H A & A & %
D ( %0
9 H? A
5? A
D = 9 9 % ( % <6 3++& I % ! % & I A
/
BA + ' % . ( 0 9 ( D
!0 . !9 ( 3 3++&
I
2 +&6 3 (
D + ' ++& Æ < ( %0
9 H5 ? A5 ? A
( +& %
$ ! .-
2 6 ( % &6 3 +6 @ & ( 30 ' 4 4//< 2 !
6 < 3+ 1 < ++& ( & &0 % & & N $ 1 6 =N .3+ <( ( & % N $' % N . & A (6 B ( % & 3 ++&
/"/
; + % & ; ? & .3++& ( ?! < 0
I 0 ? H ?I 0 ? H ?
. & ?
3+6 + <( D F 3 % & < % B
F H ;
% ? 3A< +( 3 0 3 H & ( % & % ? % 1
/"
.3 3+ + 3++& I 3
' H
F
H
F
; 3++& I6 ( 3 3 F +( ( 2 6 ' % & H & & 1 &
. 3 3% = < 6 ( 3 %
& % % & 5 5 3 & & % ; % 3 &
< 66 3++& I
3 & 3 '& & 6 & 3 1 3++& I D 3 ++&
( 3++& I 3 3 A% % 6 ' F8 = ' ! 2 6 < & (3& ( <
/"/ 54
; + % & ; ? & .3++& ( ?! < 0
I 0 ? H ? I 0 ? H ?
D ( & ?
; < ( & 3+ ( 3+ B ? .3 3 % ' ?0
' H #?
. % 5 1'6 3++& I ' ( 6 . % 6
)% ' (
/-
/"/ $ :
; + % & < .3++& ( < ! < 0
I 0 H I 0 H
D ( & <
.3 3 0
' H
F G
. D (
' H
L
#
$& % 3+
D I % & % ( % ' 5
. !/ 4 B ( )% .A ' < ( )% 3' ( 3 ( 3 3 !/ 4 ' 3 ( % . )% & 3' + ( !/ 4 D ( 6 3 +(
0 ! .
2 ( % 6 % 3+ !3 % %1 ++&
/)/ F !
; + % % 3 % & D 3 ! 3++& I
/7
. & A % 6 & A % % ; I % % A % ( A %
D % )
+ H =+ H = ( =
6 (3 3 A D H
) A & %
; 3++& I6 ) H = H ) H H .3 +( %
3 H
3 ( % % 7 H
Æ (6 1' 6 ( *!*# *
'%
& & 56 3++& I (%%%
%%%L
< & % D
/)/ &
; % & % % B 6 ; %0 6 3+ 6 3 % D 3 3+ % 3 ! < B Æ ; 3++& I 36 Æ B ! % +(
; ; D 8* Æ ;6
8* H 0;& 7 ;&7
H
!;&
" !
"
H -
& H
F;& G
; "* % 8* 3+ . 8* 3++& I !6 + =
/8
( ( 8* '%
; ( "* < ( 5 Æ ;6 3++& I6
; =6 % % 3
1 ! ."
* 36 % 3 ' +6 & % Q % .3++& 0 % & % 6 '+ % 3 3 < ++&6 6 I6 3++& % ( < ( % % 1% A
2 6 ( + 2 & A % B % 6 3% ,;+ < , +
/*/ 4 0&
! 3( (3 ' + 0
& 0 F# G & 0 F# G
D + % % % ! & ( 3++&
D (*
$&
3 % %
& ! ( ; 3++& I 3 % H 6
(
H*
*
+;
$ (6 ' ( 1 3 % % & . ( < % . % & 3 & & B +;
/:
/*/ &
! 3( & ' + B % % ( +; D %0
. %*
$% &
. +( X X 3 % & & % # # 3
$
$
. (*
*
$ & L
. % & & % # # 3
L
. % 6 % ; 1
H
$
-
**
$
3& 3 < BV &
H& & # #-
L
L
L
; 3++& I6 ( < ! + 6 % & 3 & & B % ;
< 1 ( ; 3( (3++& 3 % 3 % < ( + (( O 3 % +( +
3
.3 % +( + ! % & D 3 3A 3 < (< ' & & 8 8
/=
/+/ #
+( H 3 < '< 6 3 +
X . H
D & ( +( + 3 % & % F# G $ 6 %
I 0 # H # H H # H #I 0 1 # H #
D H # L =
& = % & %
3 D & <
H L5L= & % 5 3A % <
'<
2 & 3++& I 6 3 3 % # 5 &
/+/ #
D & ( 3
& H
H
&
. %
H
&
D < ( % % 6
% %6
= %
H L
= H
& & L
&
. % % < '< 6 %
= %
D % < % % +(+
= H
/
!+( (*
$ & 2 (
*
$
&
; 3++& I6 % & B < ( (*
$ & 6
*
$6 &
D 3
H*
*
+;
; % 3 & % ( 3 % +; & & 56 & 3? <'< 6 3++& I
/+/ 1
. 3++& I 1 ( A (3 # # % 3++& D 1 A # # ( 3 %
@ + & ;+A0 3%&
# # B
L
X X # # L B
L
% 5
H 5
& B ( 3 (
B H
=
D ( 3++& I 1% A =
. +(
X X B
L
! " # H #
H 6 C
6 &6 3 % D & < % 1 # H # # H # # H #
/
! " #$ % & #
$ 9
. ( B & ( < < ; 3 46 ( % & % .3 ( 3 % !3' ( % B 3 Æ!3 ( ' ' 1 ' 3 BV ++&
2 & ( (3 <6 % 3<6 % % 3 % < . 3 + '6 6 + < 3% D & % ' %
% + 1 ( % C
% ' ( % & ++&
. B ( ;! 0 ! ( A 3 3 + ( 3 3 !+ <0
! (
! + ( % + & 1 & 1
! ( 1 33 ;!
. ;! & 3 0
?^+C
+C
+ C
/
A C
<( +C
+C
C
C
* + % % +( % ! + % %( ( . + <
$ + .
. B ( ( ( B & %< % @ B 3 ( < A 63 + 6 3
. & B ( 0
@ A ( B
. +(
. B % 3%
. & + +& < % B
" @ < B
- . 1% B
/
"//
(3 B B6 BV ( % ! 3% (
. 0
H >*33*3$
. ! Æ 3 & 3 1 E 3% "= & == % & % 3% E? H 5 4 6 ?? H 6- 4 6 - < 3 3+
H >*3" "3*3 $
& 3 +(
H >*33*3
$,) & 4 ' %
H >*3) )3*3
$,)
* 3++&
'6 B (6 Æ B & ! Æ6 & 6 & ++& ( %1 * 3 6 ( & % D < ++& < 6 6 3 % ++& $6 % < < . B % ( '
! !( =-7 *& %=-7 *& % 4%
*& & -7 4 & , & &
$'
.3 B % 3% 3 6 1 %
/
.3 ++& B Æ .( < 6 3 ( ( 3 $ <<6 % ( 3++& 3++& . % < % $ 6 Æ 3 & R <( < Æ 3 + =6 % < @ 3( & (
"//
! ( 6 !6 <
; + % H . !+ . 1
H >*33*3>3
> H >*3>
3 H 3*33
H #
>
3
H >*33*3
-
)
H >*3) )3*3
-
)
%
#6 ( # H
# H
E ? % ' ( ::8-"P 3+6 3 &
E H L ::8-"L ="
==
? H L ::8-" L ="
==
% H %
& H %
& %& 3 &
/
! B ( + ( 3 %1 $ A6 % < - ( ::8-"P ( !
"// # %
2'
! B & % %0 $ % % < +( ! O0 6 %6 & % '& E? ?? . % % .3 < + % 2 O %6 < & +& ( O <6 % 6 .3 +( ' %
2
3
4
5
6
7
8
9
0 10 20 30 40 50 60 70 80 90 100Time
USL
LSL
T
m(x)USL(x)LSL(x)
T(x)
"0 $' B & 1 % < ( '
! B 0 D $ & & (< 6% <6 % '& .3 % % ( B
/"
(
! + (( ' 3 + % %
1
* & & % 7 + A 2 & A N
# 0 H
H =
5 ' 3 (3 %+ <EN
@ + ^ -6 6 & % A6 6 ,6 !6 2 2 & ^ %B ( ^ & ( N
; 6 %& 1 B % D & ' %' %& 0 3 H
.. 3 H
. 4
( 3 3 5 1 3%& 33N ! 3
3 + ( H =-6 H = 6 H =6 . H =6
. H =6
. H =
.. H =="
@ ( = ( 6 + B < 3 ( ' ( & (3 Æ (3 ' < ( < D ( +( & == "===& + < ( % %& ! 3 "=== &+ <6 +( < 3 6 +( < ' 6 < 3 ! < & < & =P
# 0 =8 === == == "
" D J D 5 N
# 0 H//
/
) . 3 % % % 0
= = = = = = " =
/
2 ( 9 ( ( 9 < & = N
* D ( ' % B % '6 B '6 <' '6 (3 B ' =" D ( ' ' % ' 2 ( ' ' % ' + (3 B '2 < ( ' ' B '
# 0
+ . 3 < %3A % ' < < ( 3 & ' . 0
/ 3A = & = === = === & == === L == === ! & '
== " =
! 3 3 '
== "= "=
2O + + %0 6 6 6 '6 '6 ' 6 '6 '6 ' Y %3A 3' 3 N
# 0 H =7=86 H == 6 H ==886 ' H ==86 ' H ==76 ' H ==-6 ' H =7 6 ' H =:6 ' H ==7 % 3A 3' 3 3& 3 ' '
- 3Æ 3 6 "== 6 === ; "== ; ; ;
% . %0
!
!
# ; 8= "= 77=# ; "= = =
2 + 0 (3 6 (3 (3 + (3
5 ( & 3Æ 0 < 6 &
/
@ + & < 2 : &
< ( +6 O & < & 6 % O % O % 6 ( & 3 ' &'
. (3 ( % & = ;+ (3 < " 6 O +&I < + I ' % = "
# 0 .3%& I & +I B = . 3%& I < + I 3 3 " = H
=8= H ==== " H =7-8
1 '
D ( < % 3 & & 9 H ! %& %0 < < $ 7 <
# 0 ==8 =7- =7
; < E 1 3% ( FLG 5 3 % D & & 3'( E 5 ( % %
N
2 == 6 % D ( B 3% 5 3 % N
; & % & H ! & " 2 3 ,*+ +% !
4 ( 3& H
& % #+ D (
$
H
" ; ) H
& % & 0 = ' & ' 5 ) N
2 % 3) 7 ")
# 0 16 ) 3 +( = % =
) D % 3+ 3 + % ==== % . < ="= D < (=== & .3 ( +( 6 3 =: 6 6 3 3 6
/
7"= & . ' % 3 & "== & 6 =" (3 6 = (3 =P = (3 "P2 ( %6 % < +( ( % % '
! 3 +( % & 3 '
! 3 +( +A 3A & 6 ( ' %
5 3 +( % 3 '
* . 3 & $ = 3B < & "=== 6 = 3B & "===
2 &
. B % = . + 1' 7"===2 % $ (
5 N ( ' % % 5 ( N 5 < N
# 0 H ==== := . & H "=== :== H =78: . & H 7"=== -7"-7 H =8--" 3 < ( <
+ < 6 & ' ' 6 % = &6 == ) . < &0 ) 0 = . & ) + == & % == ===" . ' &0 B H
B H
5 & 3 N
- 2O % " & ( " + ( & % 3
# 0 & ( " H " & " H & " & " 6 ) H
& ( " H ) ( ) ( H ) ( L ) (
=66 & ( " H =:77L=::8-" H=:7":
/
@ :=== 3 % B 6 6 ' 36 B % % ' & 5 ( N 5 % B ( < & P N
# 0 J " H ==" & == 6 B & :-"7
1
; & 3 % !O3 B % + ' % 5 N
!O 3 B9 & 3 + ' %
@ + < ( & & ( < ( D & & &% 3 + & ( ; +6 H =6 "P & < .3 & ' -P 5 %O% < & N
; 3+ + H % H + H "- "" 7 7 8 :: :8-=": : 2 & 3 & < + H L ' ' 5 1 & N
D 3 + % 3 D & B H
B H
B . % 6 H = B H 8= 5
( % % 3 & 8 N
" D 3 = % 3 + ( 3 & % 6 . % H = 2 3% 1 5 H =
) 3 + = 6 3 % .3 6 3 D % : 3 5 3% 1 5 H =: N
# 0 F 8G* 3 + = %6 3 % .3 6 3 % # H " H " 5 1 3 % N
/"
# 0 ( ( - H =7"8
+ @ ( +< % & 3 + < === $ %6 8P 3 <% 6 <% -P @ % I. +< % JJI !O
#0 D % 1 % :"P6 % F" G F :G % 8P -P . '% %6 (3 +< % . 3% 1
% H :- 1 =:"6 H === H =- =8
- . +A 3A 3 ]$* 3 & "= ) ; O 6 +A 3A == ) 2O % 1 % =:8
# 0 F--7C -G
1 ! .""
. -7= % %0
2 P P ::" : 8 8:" : : "7=" : "8 "=" 8 7 87" 87 : -" : "7" = "" -" : 7 7 :" "7 8" 8 " -- :== " 8 : "" - : :8""" : ==
! N
; %0 " 7 - : 8 = : ! + < N
/-
# 0 D 3 ; 6 %
" - 7 8 : = "; " 7 - : 8 = :# " 7 - : 8 " =;* " - 7 8 : : =
& % ( ; # 1 % '& ( ;* .3 ; %
H =88- 23& ;6 3++& 3 (( % (
D A & D %
; " 7 - : 8 = :+ L L L L L L L
& + A ' % % .3 A & ' H
H = D 3++& 3
(( (
; ' 6 3 (3 B 3
D 3 + "== % % % & D & & % 3 =;+ ( 3 7 6 3(
# 0 . H 7 H ' & %%
%=:: : =
=::" :
' 6
H =:: L =::" =:: = :
: : H =::
.3++& (3& ( =P ( & <
D ' + (3 & ' % &
& 6 %6 5 O% 3 ' + N
/7
D 1 +BV 3 + 3 6 < % D % % 3'0
4 % &
8 -" "" "" : "" " - "7 "8 7 ": " = :
! 3 1 +BV
" D 3 + ( % + D %0
H H -: H 8 H H "8
* & ' 1 )%;% !/ 4
) . & + < ( & & H 8# H "# . + 6 < ( B& % + D ( < B + " & 6 < 6 # H "# . + < ( + (==" N
# 0 D 3++& I 0 # H 8 % 2 ' 6 ( + <
1 4+
; +BV 6 & & H == !+ X H 7= H " . 1 ?? E? H "= 7" == ! 5 %O% N
# 0 .
/8
-7 8 8
& 6 % 3 B ( 4 < @ <
D + B < ( & & < H ==
;+ (3 &% + 6 ( % D6 & < 3+ N
2 (
D ( H =:"
D ( H =::
+ ( H ==
!O B (O 3
$ B6 ( < % ( <' H == H ==8 N 53O% N
/:
)
.3 < * ( ( 3 & 6 % O 3 (3 < = = = 7"
1 % & (6 %
. + % . & ' 6 6 6 6 6 6 6 6 6 %' ( 6 6 6 6 6 6 6 6 . 6 % <6 " D %0
% H
% H
% H
% H
% H
% H
% H
% H
! 3 %0
E7 H./.
E7 H
./.
E7 H./.
E7 H
./.
D 1
% HE7 L E7 L E7 L E7
B 6 % : +( %6 = +( % ( +( % E7 $16 % = &
! ( % %O <0
2O 3 3B 2O 3 7" 2O6 % 6 ++& ( % B
(O % & ' ' ++
/ % % &6 ++& ( % %O % ( 3 &%6 6 6 ( % ( %
! "#
T
* +
! + % % % % ( 3 (
F!+ : G !+ ` ). :: ! ! < ;+ < 2 ^+ 6 7+ 8 * 6 ;6 =a
F2+ :-G D2+% ::- 6 6 @% 6 5 6
F2 :7G $2 ::7 9 6 $ ;6 ! ;
F2 7G #D2 ` $> :7 ! (4 :+6 ]+ U` ;6 ^ Y9
FR 8G !R ` ! :8 . : 6 4
F :"G ;D ; ::" 7+ (6 + 6 ;D ; > 9
F]+ :G .]+ ` ;)O :: ! (+ 9 6 !+ ` >
F) :G 4) :: * 6 +( %
F4 :-G 2!4 ::- 9 7+ (6 6 U 6
F ==G R === ;< + ( 7+6 @% < +6 2% < 4< $ D 4
F8-G U>6 ,6 ;*99 ` U*/6 = > *: 4 (6 ! @% 6 :8-
F#:-G ]# ` R!+%6 ! ; 6 !6 ! *26 6 ::-
F; :=G R; ::= ! :+ 6 $ *+
F;+ 8 G 2;+^O :8 . 66 4;6 ! ;(
F;+ 8=G ]];+^O :8= ( !6 6 2<(6 ; .
,
F;+ 8=G ]];+^O :8= 6 62 <(6 ; .
F;+ 88G ,;+ :88 /6 $ R 4
2 B6 % % (( 3 U ! 31 % %O ( %3 <
+0QQ^^^<Q' 9Q++ 0 (% %O % (
+0QQ^^^< Q9Q'Q+ 0 ' ( 3 %
,
SY02Tables Statistiques
T. Denœux et G. Govaert
Automne 2004
Table des matieres
1 Distributions de probabilite 21.1 Fonction de repartition de la loi binomiale . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Fonction de repartition de la loi de Poisson . . . . . . . . . . . . . . . . . . . . . . . 101.3 Fonction de repartition de la loi Normale centree reduite . . . . . . . . . . . . . . . . 141.4 Fractiles de la loi Normale centree reduite . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Fractiles de la loi du χ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6 Fractiles de la loi de Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.7 Fractiles de la loi de Fisher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Intervalles de confiance pour une proportion 222.1 Intervalle bilateral (1 − α = 0.90) et intervalle unilateral (1 − α = 0.95) . . . . . . . 222.2 Intervalle bilateral (1 − α = 0.95) et intervalle unilateral (1 − α = 0.975) . . . . . . . 232.3 Intervalle bilateral (1 − α = 0.98) et intervalle unilateral (1 − α = 0.99) . . . . . . . 242.4 Intervalle bilateral (1 − α = 0.99) et intervalle unilateral (1 − α = 0.995) . . . . . . . 25
3 Puissance du test de Student 263.1 Tests bilateraux pour α = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Tests bilateraux pour α = 0.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Tests unilateraux pour α = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Tests unilateraux pour α = 0.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Test de Wilcoxon 304.1 Test bilateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Test unilateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Test de Wilcoxon signe 32
6 Distribution de Kolmogorov-Smirnov 33
7 Formulaire 34
Attention
Pour etre utilisable en examen, ce document ne doit comporteraucune surcharge manuscrite.
1
1 Distributions de probabilite
1.1 Fonction de repartition de la loi binomiale
– Si X ∼ B(n, p), alors P(X = x) = Cxnpx(1 − p)n−x∀x ∈ 1, . . . , n, E(X) = np et Var(X) =
np(1 − p).– La table qui suit donne la fonction de repartition pour les valeurs de p ≤ 0.5. Sachant que si
X ∼ B(n, p) alors n−X ∼ B(n, 1−p), on peut en deduire facilement la fonction de repartitionpour les valeurs de p superieures a 0.5.
– Enfin, pour les grandes valeurs de n, on pourra utiliser, si np et n(1 − p) sont superieurs a 5,
l’approximation gaussienne : P(X ≤ x) Φ(
x+0.5−np√np(1−p)
)ou Φ est la fonction de repartition
de la loi normale centree reduite.
P(X ≤ x) ou X ∼ B(n, p)
pn x .05 .10 .15 .20 .25 .30 .35 .40 .45 .502 0 0.9025 0.8100 0.7225 0.6400 0.5625 0.4900 0.4225 0.3600 0.3025 0.2500
1 0.9975 0.9900 0.9775 0.9600 0.9375 0.9100 0.8775 0.8400 0.7975 0.7500
3 0 0.8574 0.7290 0.6141 0.5120 0.4219 0.3430 0.2746 0.2160 0.1664 0.12501 0.9927 0.9720 0.9392 0.8960 0.8438 0.7840 0.7182 0.6480 0.5748 0.50002 0.9999 0.9990 0.9966 0.9920 0.9844 0.9730 0.9571 0.9360 0.9089 0.8750
4 0 0.8145 0.6561 0.5220 0.4096 0.3164 0.2401 0.1785 0.1296 0.0915 0.06251 0.9860 0.9477 0.8905 0.8192 0.7383 0.6517 0.5630 0.4752 0.3910 0.31252 0.9995 0.9963 0.9880 0.9728 0.9492 0.9163 0.8735 0.8208 0.7585 0.68753 1 0.9999 0.9995 0.9984 0.9961 0.9919 0.9850 0.9744 0.9590 0.9375
5 0 0.7738 0.5905 0.4437 0.3277 0.2373 0.1681 0.1160 0.0778 0.0503 0.03121 0.9774 0.9185 0.8352 0.7373 0.6328 0.5282 0.4284 0.3370 0.2562 0.18752 0.9988 0.9914 0.9734 0.9421 0.8965 0.8369 0.7648 0.6826 0.5931 0.50003 1 0.9995 0.9978 0.9933 0.9844 0.9692 0.9460 0.9130 0.8688 0.81254 1 1 0.9999 0.9997 0.9990 0.9976 0.9947 0.9898 0.9815 0.9688
6 0 0.7351 0.5314 0.3771 0.2621 0.1780 0.1176 0.0754 0.0467 0.0277 0.01561 0.9672 0.8857 0.7765 0.6554 0.5339 0.4202 0.3191 0.2333 0.1636 0.10942 0.9978 0.9842 0.9527 0.9011 0.8306 0.7443 0.6471 0.5443 0.4415 0.34383 0.9999 0.9987 0.9941 0.9830 0.9624 0.9295 0.8826 0.8208 0.7447 0.65624 1 0.9999 0.9996 0.9984 0.9954 0.9891 0.9777 0.9590 0.9308 0.8906
5 1 1 1 0.9999 0.9998 0.9993 0.9982 0.9959 0.9917 0.9844
7 0 0.6983 0.4783 0.3206 0.2097 0.1335 0.0824 0.0490 0.0280 0.0152 0.00781 0.9556 0.8503 0.7166 0.5767 0.4449 0.3294 0.2338 0.1586 0.1024 0.06252 0.9962 0.9743 0.9262 0.8520 0.7564 0.6471 0.5323 0.4199 0.3164 0.22663 0.9998 0.9973 0.9879 0.9667 0.9294 0.8740 0.8002 0.7102 0.6083 0.50004 1 0.9998 0.9988 0.9953 0.9871 0.9712 0.9444 0.9037 0.8471 0.7734
5 1 1 0.9999 0.9996 0.9987 0.9962 0.9910 0.9812 0.9643 0.93756 1 1 1 1 0.9999 0.9998 0.9994 0.9984 0.9963 0.9922
8 0 0.6634 0.4305 0.2725 0.1678 0.1001 0.0576 0.0319 0.0168 0.0084 0.00391 0.9428 0.8131 0.6572 0.5033 0.3671 0.2553 0.1691 0.1064 0.0632 0.03522 0.9942 0.9619 0.8948 0.7969 0.6785 0.5518 0.4278 0.3154 0.2201 0.14453 0.9996 0.9950 0.9786 0.9437 0.8862 0.8059 0.7064 0.5941 0.4770 0.36334 1 0.9996 0.9971 0.9896 0.9727 0.9420 0.8939 0.8263 0.7396 0.6367
5 1 1 0.9998 0.9988 0.9958 0.9887 0.9747 0.9502 0.9115 0.85556 1 1 1 0.9999 0.9996 0.9987 0.9964 0.9915 0.9819 0.96487 1 1 1 1 1 0.9999 0.9998 0.9993 0.9983 0.9961
9 0 0.6302 0.3874 0.2316 0.1342 0.0751 0.0404 0.0207 0.0101 0.0046 0.00201 0.9288 0.7748 0.5995 0.4362 0.3003 0.1960 0.1211 0.0705 0.0385 0.01952 0.9916 0.9470 0.8591 0.7382 0.6007 0.4628 0.3373 0.2318 0.1495 0.08983 0.9994 0.9917 0.9661 0.9144 0.8343 0.7297 0.6089 0.4826 0.3614 0.25394 1 0.9991 0.9944 0.9804 0.9511 0.9012 0.8283 0.7334 0.6214 0.5000
5 1 0.9999 0.9994 0.9969 0.9900 0.9747 0.9464 0.9006 0.8342 0.74616 1 1 1 0.9997 0.9987 0.9957 0.9888 0.9750 0.9502 0.91027 1 1 1 1 0.9999 0.9996 0.9986 0.9962 0.9909 0.98058 1 1 1 1 1 1 0.9999 0.9997 0.9992 0.9980
2
P(X ≤ x) ou X ∼ B(n, p)
pn x .05 .10 .15 .20 .25 .30 .35 .40 .45 .5010 0 0.5987 0.3487 0.1969 0.1074 0.0563 0.0282 0.0135 0.0060 0.0025 0.0010
1 0.9139 0.7361 0.5443 0.3758 0.2440 0.1493 0.0860 0.0464 0.0233 0.01072 0.9885 0.9298 0.8202 0.6778 0.5256 0.3828 0.2616 0.1673 0.0996 0.05473 0.9990 0.9872 0.9500 0.8791 0.7759 0.6496 0.5138 0.3823 0.2660 0.17194 0.9999 0.9984 0.9901 0.9672 0.9219 0.8497 0.7515 0.6331 0.5044 0.3770
5 1 0.9999 0.9986 0.9936 0.9803 0.9527 0.9051 0.8338 0.7384 0.62306 1 1 0.9999 0.9991 0.9965 0.9894 0.9740 0.9452 0.8980 0.82817 1 1 1 0.9999 0.9996 0.9984 0.9952 0.9877 0.9726 0.94538 1 1 1 1 1 0.9999 0.9995 0.9983 0.9955 0.98939 1 1 1 1 1 1 1 0.9999 0.9997 0.9990
11 0 0.5688 0.3138 0.1673 0.0859 0.0422 0.0198 0.0088 0.0036 0.0014 0.00051 0.8981 0.6974 0.4922 0.3221 0.1971 0.1130 0.0606 0.0302 0.0139 0.00592 0.9848 0.9104 0.7788 0.6174 0.4552 0.3127 0.2001 0.1189 0.0652 0.03273 0.9984 0.9815 0.9306 0.8389 0.7133 0.5696 0.4256 0.2963 0.1911 0.11334 0.9999 0.9972 0.9841 0.9496 0.8854 0.7897 0.6683 0.5328 0.3971 0.2744
5 1 0.9997 0.9973 0.9883 0.9657 0.9218 0.8513 0.7535 0.6331 0.50006 1 1 0.9997 0.9980 0.9924 0.9784 0.9499 0.9006 0.8262 0.72567 1 1 1 0.9998 0.9988 0.9957 0.9878 0.9707 0.9390 0.88678 1 1 1 1 0.9999 0.9994 0.9980 0.9941 0.9852 0.96739 1 1 1 1 1 1 0.9998 0.9993 0.9978 0.9941
10 1 1 1 1 1 1 1 1 0.9998 0.9995
12 0 0.5404 0.2824 0.1422 0.0687 0.0317 0.0138 0.0057 0.0022 0.0008 0.00021 0.8816 0.6590 0.4435 0.2749 0.1584 0.0850 0.0424 0.0196 0.0083 0.00322 0.9804 0.8891 0.7358 0.5583 0.3907 0.2528 0.1513 0.0834 0.0421 0.01933 0.9978 0.9744 0.9078 0.7946 0.6488 0.4925 0.3467 0.2253 0.1345 0.07304 0.9998 0.9957 0.9761 0.9274 0.8424 0.7237 0.5833 0.4382 0.3044 0.1938
5 1 0.9995 0.9954 0.9806 0.9456 0.8822 0.7873 0.6652 0.5269 0.38726 1 0.9999 0.9993 0.9961 0.9857 0.9614 0.9154 0.8418 0.7393 0.61287 1 1 0.9999 0.9994 0.9972 0.9905 0.9745 0.9427 0.8883 0.80628 1 1 1 0.9999 0.9996 0.9983 0.9944 0.9847 0.9644 0.92709 1 1 1 1 1 0.9998 0.9992 0.9972 0.9921 0.9807
10 1 1 1 1 1 1 0.9999 0.9997 0.9989 0.996811 1 1 1 1 1 1 1 1 0.9999 0.9998
13 0 0.5133 0.2542 0.1209 0.0550 0.0238 0.0097 0.0037 0.0013 0.0004 0.00011 0.8646 0.6213 0.3983 0.2336 0.1267 0.0637 0.0296 0.0126 0.0049 0.00172 0.9755 0.8661 0.6920 0.5017 0.3326 0.2025 0.1132 0.0579 0.0269 0.01123 0.9969 0.9658 0.8820 0.7473 0.5843 0.4206 0.2783 0.1686 0.0929 0.04614 0.9997 0.9935 0.9658 0.9009 0.7940 0.6543 0.5005 0.3530 0.2279 0.1334
5 1 0.9991 0.9925 0.9700 0.9198 0.8346 0.7159 0.5744 0.4268 0.29056 1 0.9999 0.9987 0.9930 0.9757 0.9376 0.8705 0.7712 0.6437 0.50007 1 1 0.9998 0.9988 0.9944 0.9818 0.9538 0.9023 0.8212 0.70958 1 1 1 0.9998 0.9990 0.9960 0.9874 0.9679 0.9302 0.86669 1 1 1 1 0.9999 0.9993 0.9975 0.9922 0.9797 0.9539
10 1 1 1 1 1 0.9999 0.9997 0.9987 0.9959 0.988811 1 1 1 1 1 1 1 0.9999 0.9995 0.998312 1 1 1 1 1 1 1 1 1 0.9999
14 0 0.4877 0.2288 0.1028 0.0440 0.0178 0.0068 0.0024 0.0008 0.0002 0.00011 0.8470 0.5846 0.3567 0.1979 0.1010 0.0475 0.0205 0.0081 0.0029 0.00092 0.9699 0.8416 0.6479 0.4481 0.2811 0.1608 0.0839 0.0398 0.0170 0.00653 0.9958 0.9559 0.8535 0.6982 0.5213 0.3552 0.2205 0.1243 0.0632 0.02874 0.9996 0.9908 0.9533 0.8702 0.7415 0.5842 0.4227 0.2793 0.1672 0.0898
5 1 0.9985 0.9885 0.9561 0.8883 0.7805 0.6405 0.4859 0.3373 0.21206 1 0.9998 0.9978 0.9884 0.9617 0.9067 0.8164 0.6925 0.5461 0.39537 1 1 0.9997 0.9976 0.9897 0.9685 0.9247 0.8499 0.7414 0.60478 1 1 1 0.9996 0.9978 0.9917 0.9757 0.9417 0.8811 0.78809 1 1 1 1 0.9997 0.9983 0.9940 0.9825 0.9574 0.9102
10 1 1 1 1 1 0.9998 0.9989 0.9961 0.9886 0.971311 1 1 1 1 1 1 0.9999 0.9994 0.9978 0.993512 1 1 1 1 1 1 1 0.9999 0.9997 0.999113 1 1 1 1 1 1 1 1 1 0.9999
3
P(X ≤ x) ou X ∼ B(n, p)
pn x .05 .10 .15 .20 .25 .30 .35 .40 .45 .5015 0 0.4633 0.2059 0.0874 0.0352 0.0134 0.0047 0.0016 0.0005 0.0001 0.0000
1 0.8290 0.5490 0.3186 0.1671 0.0802 0.0353 0.0142 0.0052 0.0017 0.00052 0.9638 0.8159 0.6042 0.3980 0.2361 0.1268 0.0617 0.0271 0.0107 0.00373 0.9945 0.9444 0.8227 0.6482 0.4613 0.2969 0.1727 0.0905 0.0424 0.01764 0.9994 0.9873 0.9383 0.8358 0.6865 0.5155 0.3519 0.2173 0.1204 0.0592
5 0.9999 0.9978 0.9832 0.9389 0.8516 0.7216 0.5643 0.4032 0.2608 0.15096 1 0.9997 0.9964 0.9819 0.9434 0.8689 0.7548 0.6098 0.4522 0.30367 1 1 0.9994 0.9958 0.9827 0.9500 0.8868 0.7869 0.6535 0.50008 1 1 0.9999 0.9992 0.9958 0.9848 0.9578 0.9050 0.8182 0.69649 1 1 1 0.9999 0.9992 0.9963 0.9876 0.9662 0.9231 0.8491
10 1 1 1 1 0.9999 0.9993 0.9972 0.9907 0.9745 0.940811 1 1 1 1 1 0.9999 0.9995 0.9981 0.9937 0.982412 1 1 1 1 1 1 0.9999 0.9997 0.9989 0.996313 1 1 1 1 1 1 1 1 0.9999 0.999514 1 1 1 1 1 1 1 1 1 1
16 0 0.4401 0.1853 0.0743 0.0281 0.0100 0.0033 0.0010 0.0003 0.0001 0.00001 0.8108 0.5147 0.2839 0.1407 0.0635 0.0261 0.0098 0.0033 0.0010 0.00032 0.9571 0.7892 0.5614 0.3518 0.1971 0.0994 0.0451 0.0183 0.0066 0.00213 0.9930 0.9316 0.7899 0.5981 0.4050 0.2459 0.1339 0.0651 0.0281 0.01064 0.9991 0.9830 0.9209 0.7982 0.6302 0.4499 0.2892 0.1666 0.0853 0.0384
5 0.9999 0.9967 0.9765 0.9183 0.8103 0.6598 0.4900 0.3288 0.1976 0.10516 1 0.9995 0.9944 0.9733 0.9204 0.8247 0.6881 0.5272 0.3660 0.22727 1 0.9999 0.9989 0.9930 0.9729 0.9256 0.8406 0.7161 0.5629 0.40188 1 1 0.9998 0.9985 0.9925 0.9743 0.9329 0.8577 0.7441 0.59829 1 1 1 0.9998 0.9984 0.9929 0.9771 0.9417 0.8759 0.7728
10 1 1 1 1 0.9997 0.9984 0.9938 0.9809 0.9514 0.894911 1 1 1 1 1 0.9997 0.9987 0.9951 0.9851 0.961612 1 1 1 1 1 1 0.9998 0.9991 0.9965 0.989413 1 1 1 1 1 1 1 0.9999 0.9994 0.997914 1 1 1 1 1 1 1 1 0.9999 0.9997
15 1 1 1 1 1 1 1 1 1 1
17 0 0.4181 0.1668 0.0631 0.0225 0.0075 0.0023 0.0007 0.0002 0.0000 0.00001 0.7922 0.4818 0.2525 0.1182 0.0501 0.0193 0.0067 0.0021 0.0006 0.00012 0.9497 0.7618 0.5198 0.3096 0.1637 0.0774 0.0327 0.0123 0.0041 0.00123 0.9912 0.9174 0.7556 0.5489 0.3530 0.2019 0.1028 0.0464 0.0184 0.00644 0.9988 0.9779 0.9013 0.7582 0.5739 0.3887 0.2348 0.1260 0.0596 0.0245
5 0.9999 0.9953 0.9681 0.8943 0.7653 0.5968 0.4197 0.2639 0.1471 0.07176 1 0.9992 0.9917 0.9623 0.8929 0.7752 0.6188 0.4478 0.2902 0.16627 1 0.9999 0.9983 0.9891 0.9598 0.8954 0.7872 0.6405 0.4743 0.31458 1 1 0.9997 0.9974 0.9876 0.9597 0.9006 0.8011 0.6626 0.50009 1 1 1 0.9995 0.9969 0.9873 0.9617 0.9081 0.8166 0.6855
10 1 1 1 0.9999 0.9994 0.9968 0.9880 0.9652 0.9174 0.833811 1 1 1 1 0.9999 0.9993 0.9970 0.9894 0.9699 0.928312 1 1 1 1 1 0.9999 0.9994 0.9975 0.9914 0.975513 1 1 1 1 1 1 0.9999 0.9995 0.9981 0.993614 1 1 1 1 1 1 1 0.9999 0.9997 0.9988
15 1 1 1 1 1 1 1 1 1 0.999916 1 1 1 1 1 1 1 1 1 1
18 0 0.3972 0.1501 0.0536 0.0180 0.0056 0.0016 0.0004 0.0001 0.0000 0.00001 0.7735 0.4503 0.2241 0.0991 0.0395 0.0142 0.0046 0.0013 0.0003 0.00012 0.9419 0.7338 0.4797 0.2713 0.1353 0.0600 0.0236 0.0082 0.0025 0.00073 0.9891 0.9018 0.7202 0.5010 0.3057 0.1646 0.0783 0.0328 0.0120 0.00384 0.9985 0.9718 0.8794 0.7164 0.5187 0.3327 0.1886 0.0942 0.0411 0.0154
5 0.9998 0.9936 0.9581 0.8671 0.7175 0.5344 0.3550 0.2088 0.1077 0.04816 1 0.9988 0.9882 0.9487 0.8610 0.7217 0.5491 0.3743 0.2258 0.11897 1 0.9998 0.9973 0.9837 0.9431 0.8593 0.7283 0.5634 0.3915 0.24038 1 1 0.9995 0.9957 0.9807 0.9404 0.8609 0.7368 0.5778 0.40739 1 1 0.9999 0.9991 0.9946 0.9790 0.9403 0.8653 0.7473 0.5927
10 1 1 1 0.9998 0.9988 0.9939 0.9788 0.9424 0.8720 0.759711 1 1 1 1 0.9998 0.9986 0.9938 0.9797 0.9463 0.881112 1 1 1 1 1 0.9997 0.9986 0.9942 0.9817 0.951913 1 1 1 1 1 1 0.9997 0.9987 0.9951 0.984614 1 1 1 1 1 1 1 0.9998 0.9990 0.9962
15 1 1 1 1 1 1 1 1 0.9999 0.999316 1 1 1 1 1 1 1 1 1 0.999917 1 1 1 1 1 1 1 1 1 1
4
P(X ≤ x) ou X ∼ B(n, p)
pn x .05 .10 .15 .20 .25 .30 .35 .40 .45 .5019 0 0.3774 0.1351 0.0456 0.0144 0.0042 0.0011 0.0003 0.0001 0.0000 0.0000
1 0.7547 0.4203 0.1985 0.0829 0.0310 0.0104 0.0031 0.0008 0.0002 0.00002 0.9335 0.7054 0.4413 0.2369 0.1113 0.0462 0.0170 0.0055 0.0015 0.00043 0.9868 0.8850 0.6841 0.4551 0.2631 0.1332 0.0591 0.0230 0.0077 0.00224 0.9980 0.9648 0.8556 0.6733 0.4654 0.2822 0.1500 0.0696 0.0280 0.0096
5 0.9998 0.9914 0.9463 0.8369 0.6678 0.4739 0.2968 0.1629 0.0777 0.03186 1 0.9983 0.9837 0.9324 0.8251 0.6655 0.4812 0.3081 0.1727 0.08357 1 0.9997 0.9959 0.9767 0.9225 0.8180 0.6656 0.4878 0.3169 0.17968 1 1 0.9992 0.9933 0.9713 0.9161 0.8145 0.6675 0.4940 0.32389 1 1 0.9999 0.9984 0.9911 0.9674 0.9125 0.8139 0.6710 0.5000
10 1 1 1 0.9997 0.9977 0.9895 0.9653 0.9115 0.8159 0.676211 1 1 1 1 0.9995 0.9972 0.9886 0.9648 0.9129 0.820412 1 1 1 1 0.9999 0.9994 0.9969 0.9884 0.9658 0.916513 1 1 1 1 1 0.9999 0.9993 0.9969 0.9891 0.968214 1 1 1 1 1 1 0.9999 0.9994 0.9972 0.9904
15 1 1 1 1 1 1 1 0.9999 0.9995 0.997816 1 1 1 1 1 1 1 1 0.9999 0.999617 1 1 1 1 1 1 1 1 1 118 1 1 1 1 1 1 1 1 1 1
20 0 0.3585 0.1216 0.0388 0.0115 0.0032 0.0008 0.0002 0.0000 0.0000 0.00001 0.7358 0.3917 0.1756 0.0692 0.0243 0.0076 0.0021 0.0005 0.0001 0.00002 0.9245 0.6769 0.4049 0.2061 0.0913 0.0355 0.0121 0.0036 0.0009 0.00023 0.9841 0.8670 0.6477 0.4114 0.2252 0.1071 0.0444 0.0160 0.0049 0.00134 0.9974 0.9568 0.8298 0.6296 0.4148 0.2375 0.1182 0.0510 0.0189 0.0059
5 0.9997 0.9887 0.9327 0.8042 0.6172 0.4164 0.2454 0.1256 0.0553 0.02076 1 0.9976 0.9781 0.9133 0.7858 0.6080 0.4166 0.2500 0.1299 0.05777 1 0.9996 0.9941 0.9679 0.8982 0.7723 0.6010 0.4159 0.2520 0.13168 1 0.9999 0.9987 0.9900 0.9591 0.8867 0.7624 0.5956 0.4143 0.25179 1 1 0.9998 0.9974 0.9861 0.9520 0.8782 0.7553 0.5914 0.4119
10 1 1 1 0.9994 0.9961 0.9829 0.9468 0.8725 0.7507 0.588111 1 1 1 0.9999 0.9991 0.9949 0.9804 0.9435 0.8692 0.748312 1 1 1 1 0.9998 0.9987 0.9940 0.9790 0.9420 0.868413 1 1 1 1 1 0.9997 0.9985 0.9935 0.9786 0.942314 1 1 1 1 1 1 0.9997 0.9984 0.9936 0.9793
15 1 1 1 1 1 1 1 0.9997 0.9985 0.994116 1 1 1 1 1 1 1 1 0.9997 0.998717 1 1 1 1 1 1 1 1 1 0.999818 1 1 1 1 1 1 1 1 1 119 1 1 1 1 1 1 1 1 1 1
21 0 0.3406 0.1094 0.0329 0.0092 0.0024 0.0006 0.0001 0.0000 0.0000 0.00001 0.7170 0.3647 0.1550 0.0576 0.0190 0.0056 0.0014 0.0003 0.0001 0.00002 0.9151 0.6484 0.3705 0.1787 0.0745 0.0271 0.0086 0.0024 0.0006 0.00013 0.9811 0.8480 0.6113 0.3704 0.1917 0.0856 0.0331 0.0110 0.0031 0.00074 0.9968 0.9478 0.8025 0.5860 0.3674 0.1984 0.0924 0.0370 0.0126 0.0036
5 0.9996 0.9856 0.9173 0.7693 0.5666 0.3627 0.2009 0.0957 0.0389 0.01336 1 0.9967 0.9713 0.8915 0.7436 0.5505 0.3567 0.2002 0.0964 0.03927 1 0.9994 0.9917 0.9569 0.8701 0.7230 0.5365 0.3495 0.1971 0.09468 1 0.9999 0.9980 0.9856 0.9439 0.8523 0.7059 0.5237 0.3413 0.19179 1 1 0.9996 0.9959 0.9794 0.9324 0.8377 0.6914 0.5117 0.3318
10 1 1 0.9999 0.9990 0.9936 0.9736 0.9228 0.8256 0.6790 0.500011 1 1 1 0.9998 0.9983 0.9913 0.9687 0.9151 0.8159 0.668212 1 1 1 1 0.9996 0.9976 0.9892 0.9648 0.9092 0.808313 1 1 1 1 0.9999 0.9994 0.9969 0.9877 0.9621 0.905414 1 1 1 1 1 0.9999 0.9993 0.9964 0.9868 0.9608
15 1 1 1 1 1 1 0.9999 0.9992 0.9963 0.986716 1 1 1 1 1 1 1 0.9998 0.9992 0.996417 1 1 1 1 1 1 1 1 0.9999 0.999318 1 1 1 1 1 1 1 1 1 0.999919 1 1 1 1 1 1 1 1 1 1
20 1 1 1 1 1 1 1 1 1 1
5
P(X ≤ x) ou X ∼ B(n, p)
pn x .05 .10 .15 .20 .25 .30 .35 .40 .45 .5022 0 0.3235 0.0985 0.0280 0.0074 0.0018 0.0004 0.0001 0.0000 0.0000 0.0000
1 0.6982 0.3392 0.1367 0.0480 0.0149 0.0041 0.0010 0.0002 0.0000 0.00002 0.9052 0.6200 0.3382 0.1545 0.0606 0.0207 0.0061 0.0016 0.0003 0.00013 0.9778 0.8281 0.5752 0.3320 0.1624 0.0681 0.0245 0.0076 0.0020 0.00044 0.9960 0.9379 0.7738 0.5429 0.3235 0.1645 0.0716 0.0266 0.0083 0.0022
5 0.9994 0.9818 0.9001 0.7326 0.5168 0.3134 0.1629 0.0722 0.0271 0.00856 0.9999 0.9956 0.9632 0.8670 0.6994 0.4942 0.3022 0.1584 0.0705 0.02627 1 0.9991 0.9886 0.9439 0.8385 0.6713 0.4736 0.2898 0.1518 0.06698 1 0.9999 0.9970 0.9799 0.9254 0.8135 0.6466 0.4540 0.2764 0.14319 1 1 0.9993 0.9939 0.9705 0.9084 0.7916 0.6244 0.4350 0.2617
10 1 1 0.9999 0.9984 0.9900 0.9613 0.8930 0.7720 0.6037 0.415911 1 1 1 0.9997 0.9971 0.9860 0.9526 0.8793 0.7543 0.584112 1 1 1 0.9999 0.9993 0.9957 0.9820 0.9449 0.8672 0.738313 1 1 1 1 0.9999 0.9989 0.9942 0.9785 0.9383 0.856914 1 1 1 1 1 0.9998 0.9984 0.9930 0.9757 0.9331
15 1 1 1 1 1 1 0.9997 0.9981 0.9920 0.973816 1 1 1 1 1 1 0.9999 0.9996 0.9979 0.991517 1 1 1 1 1 1 1 0.9999 0.9995 0.997818 1 1 1 1 1 1 1 1 0.9999 0.999619 1 1 1 1 1 1 1 1 1 0.9999
20 1 1 1 1 1 1 1 1 1 121 1 1 1 1 1 1 1 1 1 1
23 0 0.3074 0.0886 0.0238 0.0059 0.0013 0.0003 0.0000 0.0000 0.0000 0.00001 0.6794 0.3151 0.1204 0.0398 0.0116 0.0030 0.0007 0.0001 0.0000 0.00002 0.8948 0.5920 0.3080 0.1332 0.0492 0.0157 0.0043 0.0010 0.0002 0.00003 0.9742 0.8073 0.5396 0.2965 0.1370 0.0538 0.0181 0.0052 0.0012 0.00024 0.9951 0.9269 0.7440 0.5007 0.2832 0.1356 0.0551 0.0190 0.0055 0.0013
5 0.9992 0.9774 0.8811 0.6947 0.4685 0.2688 0.1309 0.0540 0.0186 0.00536 0.9999 0.9942 0.9537 0.8402 0.6537 0.4399 0.2534 0.1240 0.0510 0.01737 1 0.9988 0.9848 0.9285 0.8037 0.6181 0.4136 0.2373 0.1152 0.04668 1 0.9998 0.9958 0.9727 0.9037 0.7709 0.5860 0.3884 0.2203 0.10509 1 1 0.9990 0.9911 0.9592 0.8799 0.7408 0.5562 0.3636 0.2024
10 1 1 0.9998 0.9975 0.9851 0.9454 0.8575 0.7129 0.5278 0.338811 1 1 1 0.9994 0.9954 0.9786 0.9318 0.8364 0.6865 0.500012 1 1 1 0.9999 0.9988 0.9928 0.9717 0.9187 0.8164 0.661213 1 1 1 1 0.9997 0.9979 0.9900 0.9651 0.9063 0.797614 1 1 1 1 0.9999 0.9995 0.9970 0.9872 0.9589 0.8950
15 1 1 1 1 1 0.9999 0.9992 0.9960 0.9847 0.953416 1 1 1 1 1 1 0.9998 0.9990 0.9952 0.982717 1 1 1 1 1 1 1 0.9998 0.9988 0.994718 1 1 1 1 1 1 1 1 0.9998 0.998719 1 1 1 1 1 1 1 1 1 0.9998
20 1 1 1 1 1 1 1 1 1 121 1 1 1 1 1 1 1 1 1 122 1 1 1 1 1 1 1 1 1 1
24 0 0.2920 0.0798 0.0202 0.0047 0.0010 0.0002 0.0000 0.0000 0.0000 0.00001 0.6608 0.2925 0.1059 0.0331 0.0090 0.0022 0.0005 0.0001 0.0000 0.00002 0.8841 0.5643 0.2798 0.1145 0.0398 0.0119 0.0030 0.0007 0.0001 0.00003 0.9702 0.7857 0.5049 0.2639 0.1150 0.0424 0.0133 0.0035 0.0008 0.00014 0.9940 0.9149 0.7134 0.4599 0.2466 0.1111 0.0422 0.0134 0.0036 0.0008
5 0.9990 0.9723 0.8606 0.6559 0.4222 0.2288 0.1044 0.0400 0.0127 0.00336 0.9999 0.9925 0.9428 0.8111 0.6074 0.3886 0.2106 0.0960 0.0364 0.01137 1 0.9983 0.9801 0.9108 0.7662 0.5647 0.3575 0.1919 0.0863 0.03208 1 0.9997 0.9941 0.9638 0.8787 0.7250 0.5257 0.3279 0.1730 0.07589 1 0.9999 0.9985 0.9874 0.9453 0.8472 0.6866 0.4891 0.2991 0.1537
10 1 1 0.9997 0.9962 0.9787 0.9258 0.8167 0.6502 0.4539 0.270611 1 1 0.9999 0.9990 0.9928 0.9686 0.9058 0.7870 0.6151 0.419412 1 1 1 0.9998 0.9979 0.9885 0.9577 0.8857 0.7580 0.580613 1 1 1 1 0.9995 0.9964 0.9836 0.9465 0.8659 0.729414 1 1 1 1 0.9999 0.9990 0.9945 0.9783 0.9352 0.8463
15 1 1 1 1 1 0.9998 0.9984 0.9925 0.9731 0.924216 1 1 1 1 1 1 0.9996 0.9978 0.9905 0.968017 1 1 1 1 1 1 0.9999 0.9995 0.9972 0.988718 1 1 1 1 1 1 1 0.9999 0.9993 0.996719 1 1 1 1 1 1 1 1 0.9999 0.9992
20 1 1 1 1 1 1 1 1 1 0.999921 1 1 1 1 1 1 1 1 1 122 1 1 1 1 1 1 1 1 1 123 1 1 1 1 1 1 1 1 1 1
6
P(X ≤ x) ou X ∼ B(n, p)
pn x .05 .10 .15 .20 .25 .30 .35 .40 .45 .5025 0 0.2774 0.0718 0.0172 0.0038 0.0008 0.0001 0.0000 0.0000 0.0000 0.0000
1 0.6424 0.2712 0.0931 0.0274 0.0070 0.0016 0.0003 0.0001 0.0000 0.00002 0.8729 0.5371 0.2537 0.0982 0.0321 0.0090 0.0021 0.0004 0.0001 0.00003 0.9659 0.7636 0.4711 0.2340 0.0962 0.0332 0.0097 0.0024 0.0005 0.00014 0.9928 0.9020 0.6821 0.4207 0.2137 0.0905 0.0320 0.0095 0.0023 0.0005
5 0.9988 0.9666 0.8385 0.6167 0.3783 0.1935 0.0826 0.0294 0.0086 0.00206 0.9998 0.9905 0.9305 0.7800 0.5611 0.3407 0.1734 0.0736 0.0258 0.00737 1 0.9977 0.9745 0.8909 0.7265 0.5118 0.3061 0.1536 0.0639 0.02168 1 0.9995 0.9920 0.9532 0.8506 0.6769 0.4668 0.2735 0.1340 0.05399 1 0.9999 0.9979 0.9827 0.9287 0.8106 0.6303 0.4246 0.2424 0.1148
10 1 1 0.9995 0.9944 0.9703 0.9022 0.7712 0.5858 0.3843 0.212211 1 1 0.9999 0.9985 0.9893 0.9558 0.8746 0.7323 0.5426 0.345012 1 1 1 0.9996 0.9966 0.9825 0.9396 0.8462 0.6937 0.500013 1 1 1 0.9999 0.9991 0.9940 0.9745 0.9222 0.8173 0.655014 1 1 1 1 0.9998 0.9982 0.9907 0.9656 0.9040 0.7878
15 1 1 1 1 1 0.9995 0.9971 0.9868 0.9560 0.885216 1 1 1 1 1 0.9999 0.9992 0.9957 0.9826 0.946117 1 1 1 1 1 1 0.9998 0.9988 0.9942 0.978418 1 1 1 1 1 1 1 0.9997 0.9984 0.992719 1 1 1 1 1 1 1 0.9999 0.9996 0.9980
20 1 1 1 1 1 1 1 1 0.9999 0.999521 1 1 1 1 1 1 1 1 1 0.999922 1 1 1 1 1 1 1 1 1 123 1 1 1 1 1 1 1 1 1 124 1 1 1 1 1 1 1 1 1 1
30 0 0.2146 0.0424 0.0076 0.0012 0.0002 0.0000 0.0000 0.0000 0.0000 0.00001 0.5535 0.1837 0.0480 0.0105 0.0020 0.0003 0.0000 0.0000 0.0000 0.00002 0.8122 0.4114 0.1514 0.0442 0.0106 0.0021 0.0003 0.0000 0.0000 0.00003 0.9392 0.6474 0.3217 0.1227 0.0374 0.0093 0.0019 0.0003 0.0000 0.00004 0.9844 0.8245 0.5245 0.2552 0.0979 0.0302 0.0075 0.0015 0.0002 0.0000
5 0.9967 0.9268 0.7106 0.4275 0.2026 0.0766 0.0233 0.0057 0.0011 0.00026 0.9994 0.9742 0.8474 0.6070 0.3481 0.1595 0.0586 0.0172 0.0040 0.00077 0.9999 0.9922 0.9302 0.7608 0.5143 0.2814 0.1238 0.0435 0.0121 0.00268 1 0.9980 0.9722 0.8713 0.6736 0.4315 0.2247 0.0940 0.0312 0.00819 1 0.9995 0.9903 0.9389 0.8034 0.5888 0.3575 0.1763 0.0694 0.0214
10 1 0.9999 0.9971 0.9744 0.8943 0.7304 0.5078 0.2915 0.1350 0.049411 1 1 0.9992 0.9905 0.9493 0.8407 0.6548 0.4311 0.2327 0.100212 1 1 0.9998 0.9969 0.9784 0.9155 0.7802 0.5785 0.3592 0.180813 1 1 1 0.9991 0.9918 0.9599 0.8737 0.7145 0.5025 0.292314 1 1 1 0.9998 0.9973 0.9831 0.9348 0.8246 0.6448 0.4278
15 1 1 1 0.9999 0.9992 0.9936 0.9699 0.9029 0.7691 0.572216 1 1 1 1 0.9998 0.9979 0.9876 0.9519 0.8644 0.707717 1 1 1 1 0.9999 0.9994 0.9955 0.9788 0.9286 0.819218 1 1 1 1 1 0.9998 0.9986 0.9917 0.9666 0.899819 1 1 1 1 1 1 0.9996 0.9971 0.9862 0.9506
20 1 1 1 1 1 1 0.9999 0.9991 0.9950 0.978621 1 1 1 1 1 1 1 0.9998 0.9984 0.991922 1 1 1 1 1 1 1 1 0.9996 0.997423 1 1 1 1 1 1 1 1 0.9999 0.999324 1 1 1 1 1 1 1 1 1 0.9998
25 1 1 1 1 1 1 1 1 1 1a 1 1 1 1 1 1 1 1 1 129 1 1 1 1 1 1 1 1 1 1
7
P(X ≤ x) ou X ∼ B(n, p)
pn x .05 .10 .15 .20 .25 .30 .35 .40 .45 .5035 0 0.1661 0.0250 0.0034 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1 0.4720 0.1224 0.0243 0.0040 0.0005 0.0001 0.0000 0.0000 0.0000 0.00002 0.7458 0.3063 0.0870 0.0190 0.0033 0.0005 0.0001 0.0000 0.0000 0.00003 0.9042 0.5310 0.2088 0.0605 0.0136 0.0024 0.0003 0.0000 0.0000 0.00004 0.9710 0.7307 0.3807 0.1435 0.0410 0.0091 0.0016 0.0002 0.0000 0.0000
5 0.9927 0.8684 0.5689 0.2721 0.0976 0.0269 0.0058 0.0010 0.0001 0.00006 0.9985 0.9448 0.7348 0.4328 0.1920 0.0650 0.0170 0.0034 0.0005 0.00017 0.9997 0.9800 0.8562 0.5993 0.3223 0.1326 0.0419 0.0102 0.0019 0.00038 1 0.9937 0.9311 0.7450 0.4743 0.2341 0.0890 0.0260 0.0057 0.00099 1 0.9983 0.9708 0.8543 0.6263 0.3646 0.1651 0.0575 0.0152 0.0030
10 1 0.9996 0.9890 0.9253 0.7581 0.5100 0.2716 0.1123 0.0354 0.008311 1 0.9999 0.9963 0.9656 0.8579 0.6516 0.4019 0.1952 0.0729 0.020512 1 1 0.9989 0.9858 0.9244 0.7729 0.5423 0.3057 0.1344 0.044813 1 1 0.9997 0.9947 0.9637 0.8650 0.6760 0.4361 0.2233 0.087714 1 1 0.9999 0.9982 0.9842 0.9269 0.7891 0.5728 0.3376 0.1553
15 1 1 1 0.9995 0.9938 0.9641 0.8744 0.7003 0.4685 0.249816 1 1 1 0.9999 0.9978 0.9840 0.9318 0.8065 0.6024 0.367917 1 1 1 1 0.9993 0.9936 0.9664 0.8857 0.7249 0.500018 1 1 1 1 0.9998 0.9977 0.9850 0.9385 0.8251 0.632119 1 1 1 1 0.9999 0.9992 0.9939 0.9700 0.8984 0.7502
20 1 1 1 1 1 0.9998 0.9978 0.9867 0.9464 0.844721 1 1 1 1 1 0.9999 0.9993 0.9947 0.9745 0.912322 1 1 1 1 1 1 0.9998 0.9981 0.9891 0.955223 1 1 1 1 1 1 0.9999 0.9994 0.9958 0.979524 1 1 1 1 1 1 1 0.9998 0.9986 0.9917
25 1 1 1 1 1 1 1 1 0.9996 0.997026 1 1 1 1 1 1 1 1 0.9999 0.999127 1 1 1 1 1 1 1 1 1 0.999728 1 1 1 1 1 1 1 1 1 0.999929 1 1 1 1 1 1 1 1 1 1a 1 1 1 1 1 1 1 1 1 134 1 1 1 1 1 1 1 1 1 1
40 0 0.1285 0.0148 0.0015 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.00001 0.3991 0.0805 0.0121 0.0015 0.0001 0.0000 0.0000 0.0000 0.0000 0.00002 0.6767 0.2228 0.0486 0.0079 0.0010 0.0001 0.0000 0.0000 0.0000 0.00003 0.8619 0.4231 0.1302 0.0285 0.0047 0.0006 0.0001 0.0000 0.0000 0.00004 0.9520 0.6290 0.2633 0.0759 0.0160 0.0026 0.0003 0.0000 0.0000 0.0000
5 0.9861 0.7937 0.4325 0.1613 0.0433 0.0086 0.0013 0.0001 0.0000 0.00006 0.9966 0.9005 0.6067 0.2859 0.0962 0.0238 0.0044 0.0006 0.0001 0.00007 0.9993 0.9581 0.7559 0.4371 0.1820 0.0553 0.0124 0.0021 0.0002 0.00008 0.9999 0.9845 0.8646 0.5931 0.2998 0.1110 0.0303 0.0061 0.0009 0.00019 1 0.9949 0.9328 0.7318 0.4395 0.1959 0.0644 0.0156 0.0027 0.0003
10 1 0.9985 0.9701 0.8392 0.5839 0.3087 0.1215 0.0352 0.0074 0.001111 1 0.9996 0.9880 0.9125 0.7151 0.4406 0.2053 0.0709 0.0179 0.003212 1 0.9999 0.9957 0.9568 0.8209 0.5772 0.3143 0.1285 0.0386 0.008313 1 1 0.9986 0.9806 0.8968 0.7032 0.4408 0.2112 0.0751 0.019214 1 1 0.9996 0.9921 0.9456 0.8074 0.5721 0.3174 0.1326 0.0403
15 1 1 0.9999 0.9971 0.9738 0.8849 0.6946 0.4402 0.2142 0.076916 1 1 1 0.9990 0.9884 0.9367 0.7978 0.5681 0.3185 0.134117 1 1 1 0.9997 0.9953 0.9680 0.8761 0.6885 0.4391 0.214818 1 1 1 0.9999 0.9983 0.9852 0.9301 0.7911 0.5651 0.317919 1 1 1 1 0.9994 0.9937 0.9637 0.8702 0.6844 0.4373
20 1 1 1 1 0.9998 0.9976 0.9827 0.9256 0.7870 0.562721 1 1 1 1 1 0.9991 0.9925 0.9608 0.8669 0.682122 1 1 1 1 1 0.9997 0.9970 0.9811 0.9233 0.785223 1 1 1 1 1 0.9999 0.9989 0.9917 0.9595 0.865924 1 1 1 1 1 1 0.9996 0.9966 0.9804 0.9231
25 1 1 1 1 1 1 0.9999 0.9988 0.9914 0.959726 1 1 1 1 1 1 1 0.9996 0.9966 0.980827 1 1 1 1 1 1 1 0.9999 0.9988 0.991728 1 1 1 1 1 1 1 1 0.9996 0.996829 1 1 1 1 1 1 1 1 0.9999 0.9989
30 1 1 1 1 1 1 1 1 1 0.999731 1 1 1 1 1 1 1 1 1 0.999932 1 1 1 1 1 1 1 1 1 1a 1 1 1 1 1 1 1 1 1 139 1 1 1 1 1 1 1 1 1 1
8
P(X ≤ x) ou X ∼ B(n, p)
pn x .05 .10 .15 .20 .25 .30 .35 .40 .45 .5045 0 0.0994 0.0087 0.0007 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1 0.3350 0.0524 0.0060 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 0.00002 0.6077 0.1590 0.0265 0.0032 0.0003 0.0000 0.0000 0.0000 0.0000 0.00003 0.8134 0.3289 0.0785 0.0129 0.0016 0.0001 0.0000 0.0000 0.0000 0.00004 0.9271 0.5271 0.1748 0.0382 0.0059 0.0007 0.0001 0.0000 0.0000 0.0000
5 0.9761 0.7077 0.3142 0.0902 0.0179 0.0026 0.0003 0.0000 0.0000 0.00006 0.9934 0.8415 0.4782 0.1768 0.0446 0.0080 0.0010 0.0001 0.0000 0.00007 0.9984 0.9243 0.6394 0.2975 0.0941 0.0209 0.0033 0.0004 0.0000 0.00008 0.9997 0.9680 0.7745 0.4407 0.1725 0.0471 0.0091 0.0012 0.0001 0.00009 0.9999 0.9880 0.8726 0.5880 0.2800 0.0934 0.0220 0.0036 0.0004 0.0000
10 1 0.9960 0.9349 0.7205 0.4089 0.1647 0.0469 0.0094 0.0013 0.000111 1 0.9988 0.9698 0.8259 0.5457 0.2620 0.0896 0.0216 0.0036 0.000412 1 0.9997 0.9873 0.9005 0.6748 0.3802 0.1547 0.0446 0.0090 0.001213 1 0.9999 0.9952 0.9479 0.7841 0.5088 0.2437 0.0836 0.0201 0.003314 1 1 0.9983 0.9750 0.8673 0.6347 0.3533 0.1430 0.0409 0.0080
15 1 1 0.9995 0.9890 0.9247 0.7462 0.4752 0.2249 0.0762 0.017816 1 1 0.9998 0.9956 0.9605 0.8358 0.5983 0.3272 0.1302 0.036217 1 1 1 0.9983 0.9809 0.9014 0.7113 0.4436 0.2056 0.067618 1 1 1 0.9994 0.9915 0.9451 0.8060 0.5643 0.3015 0.116319 1 1 1 0.9998 0.9965 0.9717 0.8785 0.6786 0.4131 0.1856
20 1 1 1 0.9999 0.9987 0.9865 0.9292 0.7777 0.5318 0.275721 1 1 1 1 0.9995 0.9940 0.9618 0.8564 0.6474 0.383022 1 1 1 1 0.9999 0.9976 0.9809 0.9135 0.7506 0.500023 1 1 1 1 1 0.9991 0.9911 0.9517 0.8350 0.617024 1 1 1 1 1 0.9997 0.9962 0.9750 0.8983 0.7243
25 1 1 1 1 1 0.9999 0.9985 0.9880 0.9418 0.814426 1 1 1 1 1 1 0.9995 0.9947 0.9692 0.883727 1 1 1 1 1 1 0.9998 0.9979 0.9850 0.932428 1 1 1 1 1 1 0.9999 0.9992 0.9932 0.963829 1 1 1 1 1 1 1 0.9997 0.9972 0.9822
30 1 1 1 1 1 1 1 0.9999 0.9990 0.992031 1 1 1 1 1 1 1 1 0.9996 0.996732 1 1 1 1 1 1 1 1 0.9999 0.998833 1 1 1 1 1 1 1 1 1 0.999634 1 1 1 1 1 1 1 1 1 0.9999
35 1 1 1 1 1 1 1 1 1 1a 1 1 1 1 1 1 1 1 1 144 1 1 1 1 1 1 1 1 1 1
9
1.2 Fonction de repartition de la loi de Poisson
Si X ∼ P(λ), alors P(X = x) = e−λ λx
x! pour x ∈ N, E(X) = λ et Var(X) = λ. La table qui suitdonne la fonction de repartition pour des valeurs de λ allant de 0 a 20. Pour les valeurs superieuresa 20, on pourra utiliser l’approximation (grossiere) gaussienne : P(X ≤ x) Φ
(x+0.5−λ√
λ
)ou Φ est
la fonction de repartition de la loi normale centree reduite.
P(X ≤ x) ou X ∼ P(λ)
λx 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00 0.9048 0.8187 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066 0.36791 0.9953 0.9825 0.9631 0.9384 0.9098 0.8781 0.8442 0.8088 0.7725 0.73582 0.9998 0.9989 0.9964 0.9921 0.9856 0.9769 0.9659 0.9526 0.9371 0.91973 1 0.9999 0.9997 0.9992 0.9982 0.9966 0.9942 0.9909 0.9865 0.98104 1 1 1 0.9999 0.9998 0.9996 0.9992 0.9986 0.9977 0.9963
5 1 1 1 1 1 1 0.9999 0.9998 0.9997 0.99946 1 1 1 1 1 1 1 1 1 0.99997 1 1 1 1 1 1 1 1 1 1
P(X ≤ x) ou X ∼ P(λ)
λx 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.00 0.3329 0.3012 0.2725 0.2466 0.2231 0.2019 0.1827 0.1653 0.1496 0.13531 0.6990 0.6626 0.6268 0.5918 0.5578 0.5249 0.4932 0.4628 0.4337 0.40602 0.9004 0.8795 0.8571 0.8335 0.8088 0.7834 0.7572 0.7306 0.7037 0.67673 0.9743 0.9662 0.9569 0.9463 0.9344 0.9212 0.9068 0.8913 0.8747 0.85714 0.9946 0.9923 0.9893 0.9857 0.9814 0.9763 0.9704 0.9636 0.9559 0.9473
5 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.9920 0.9896 0.9868 0.98346 0.9999 0.9997 0.9996 0.9994 0.9991 0.9987 0.9981 0.9974 0.9966 0.99557 1 1 0.9999 0.9999 0.9998 0.9997 0.9996 0.9994 0.9992 0.99898 1 1 1 1 1 1 0.9999 0.9999 0.9998 0.99989 1 1 1 1 1 1 1 1 1 1
P(X ≤ x) ou X ∼ P(λ)
λx 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.00 0.1225 0.1108 0.1003 0.0907 0.0821 0.0743 0.0672 0.0608 0.0550 0.04981 0.3796 0.3546 0.3309 0.3084 0.2873 0.2674 0.2487 0.2311 0.2146 0.19912 0.6496 0.6227 0.5960 0.5697 0.5438 0.5184 0.4936 0.4695 0.4460 0.42323 0.8386 0.8194 0.7993 0.7787 0.7576 0.7360 0.7141 0.6919 0.6696 0.64724 0.9379 0.9275 0.9162 0.9041 0.8912 0.8774 0.8629 0.8477 0.8318 0.8153
5 0.9796 0.9751 0.9700 0.9643 0.9580 0.9510 0.9433 0.9349 0.9258 0.91616 0.9941 0.9925 0.9906 0.9884 0.9858 0.9828 0.9794 0.9756 0.9713 0.96657 0.9985 0.9980 0.9974 0.9967 0.9958 0.9947 0.9934 0.9919 0.9901 0.98818 0.9997 0.9995 0.9994 0.9991 0.9989 0.9985 0.9981 0.9976 0.9969 0.99629 0.9999 0.9999 0.9999 0.9998 0.9997 0.9996 0.9995 0.9993 0.9991 0.9989
10 1 1 1 1 0.9999 0.9999 0.9999 0.9998 0.9998 0.999711 1 1 1 1 1 1 1 1 0.9999 0.999912 1 1 1 1 1 1 1 1 1 1
P(X ≤ x) ou X ∼ P(λ)
λx 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.00 0.0450 0.0408 0.0369 0.0334 0.0302 0.0273 0.0247 0.0224 0.0202 0.01831 0.1847 0.1712 0.1586 0.1468 0.1359 0.1257 0.1162 0.1074 0.0992 0.09162 0.4012 0.3799 0.3594 0.3397 0.3208 0.3027 0.2854 0.2689 0.2531 0.23813 0.6248 0.6025 0.5803 0.5584 0.5366 0.5152 0.4942 0.4735 0.4532 0.43354 0.7982 0.7806 0.7626 0.7442 0.7254 0.7064 0.6872 0.6678 0.6484 0.6288
5 0.9057 0.8946 0.8829 0.8705 0.8576 0.8441 0.8301 0.8156 0.8006 0.78516 0.9612 0.9554 0.9490 0.9421 0.9347 0.9267 0.9182 0.9091 0.8995 0.88937 0.9858 0.9832 0.9802 0.9769 0.9733 0.9692 0.9648 0.9599 0.9546 0.94898 0.9953 0.9943 0.9931 0.9917 0.9901 0.9883 0.9863 0.9840 0.9815 0.97869 0.9986 0.9982 0.9978 0.9973 0.9967 0.9960 0.9952 0.9942 0.9931 0.9919
10 0.9996 0.9995 0.9994 0.9992 0.9990 0.9987 0.9984 0.9981 0.9977 0.997211 0.9999 0.9999 0.9998 0.9998 0.9997 0.9996 0.9995 0.9994 0.9993 0.999112 1 1 1 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.999713 1 1 1 1 1 1 1 1 0.9999 0.999914 1 1 1 1 1 1 1 1 1 1
10
P(X ≤ x) ou X ∼ P(λ)
λx 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.00 0.0166 0.0150 0.0136 0.0123 0.0111 0.0101 0.0091 0.0082 0.0074 0.00671 0.0845 0.0780 0.0719 0.0663 0.0611 0.0563 0.0518 0.0477 0.0439 0.04042 0.2238 0.2102 0.1974 0.1851 0.1736 0.1626 0.1523 0.1425 0.1333 0.12473 0.4142 0.3954 0.3772 0.3594 0.3423 0.3257 0.3097 0.2942 0.2793 0.26504 0.6093 0.5898 0.5704 0.5512 0.5321 0.5132 0.4946 0.4763 0.4582 0.4405
5 0.7693 0.7531 0.7367 0.7199 0.7029 0.6858 0.6684 0.6510 0.6335 0.61606 0.8786 0.8675 0.8558 0.8436 0.8311 0.8180 0.8046 0.7908 0.7767 0.76227 0.9427 0.9361 0.9290 0.9214 0.9134 0.9049 0.8960 0.8867 0.8769 0.86668 0.9755 0.9721 0.9683 0.9642 0.9597 0.9549 0.9497 0.9442 0.9382 0.93199 0.9905 0.9889 0.9871 0.9851 0.9829 0.9805 0.9778 0.9749 0.9717 0.9682
10 0.9966 0.9959 0.9952 0.9943 0.9933 0.9922 0.9910 0.9896 0.9880 0.986311 0.9989 0.9986 0.9983 0.9980 0.9976 0.9971 0.9966 0.9960 0.9953 0.994512 0.9997 0.9996 0.9995 0.9993 0.9992 0.9990 0.9988 0.9986 0.9983 0.998013 0.9999 0.9999 0.9998 0.9998 0.9997 0.9997 0.9996 0.9995 0.9994 0.999314 1 1 1 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998
15 1 1 1 1 1 1 1 1 0.9999 0.999916 1 1 1 1 1 1 1 1 1 1
P(X ≤ x) ou X ∼ P(λ)
λx 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.00 0.0061 0.0055 0.0050 0.0045 0.0041 0.0037 0.0033 0.0030 0.0027 0.00251 0.0372 0.0342 0.0314 0.0289 0.0266 0.0244 0.0224 0.0206 0.0189 0.01742 0.1165 0.1088 0.1016 0.0948 0.0884 0.0824 0.0768 0.0715 0.0666 0.06203 0.2513 0.2381 0.2254 0.2133 0.2017 0.1906 0.1800 0.1700 0.1604 0.15124 0.4231 0.4061 0.3895 0.3733 0.3575 0.3422 0.3272 0.3127 0.2987 0.2851
5 0.5984 0.5809 0.5635 0.5461 0.5289 0.5119 0.4950 0.4783 0.4619 0.44576 0.7474 0.7324 0.7171 0.7017 0.6860 0.6703 0.6544 0.6384 0.6224 0.60637 0.8560 0.8449 0.8335 0.8217 0.8095 0.7970 0.7841 0.7710 0.7576 0.74408 0.9252 0.9181 0.9106 0.9027 0.8944 0.8857 0.8766 0.8672 0.8574 0.84729 0.9644 0.9603 0.9559 0.9512 0.9462 0.9409 0.9352 0.9292 0.9228 0.9161
10 0.9844 0.9823 0.9800 0.9775 0.9747 0.9718 0.9686 0.9651 0.9614 0.957411 0.9937 0.9927 0.9916 0.9904 0.9890 0.9875 0.9859 0.9841 0.9821 0.979912 0.9976 0.9972 0.9967 0.9962 0.9955 0.9949 0.9941 0.9932 0.9922 0.991213 0.9992 0.9990 0.9988 0.9986 0.9983 0.9980 0.9977 0.9973 0.9969 0.996414 0.9997 0.9997 0.9996 0.9995 0.9994 0.9993 0.9991 0.9990 0.9988 0.9986
15 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997 0.9996 0.9996 0.999516 1 1 1 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.999817 1 1 1 1 1 1 1 1 1 0.999918 1 1 1 1 1 1 1 1 1 1
P(X ≤ x) ou X ∼ P(λ)
λx 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.00 0.0022 0.0020 0.0018 0.0017 0.0015 0.0014 0.0012 0.0011 0.0010 0.00091 0.0159 0.0146 0.0134 0.0123 0.0113 0.0103 0.0095 0.0087 0.0080 0.00732 0.0577 0.0536 0.0498 0.0463 0.0430 0.0400 0.0371 0.0344 0.0320 0.02963 0.1425 0.1342 0.1264 0.1189 0.1118 0.1052 0.0988 0.0928 0.0871 0.08184 0.2719 0.2592 0.2469 0.2351 0.2237 0.2127 0.2022 0.1920 0.1823 0.1730
5 0.4298 0.4141 0.3988 0.3837 0.3690 0.3547 0.3406 0.3270 0.3137 0.30076 0.5902 0.5742 0.5582 0.5423 0.5265 0.5108 0.4953 0.4799 0.4647 0.44977 0.7301 0.7160 0.7017 0.6873 0.6728 0.6581 0.6433 0.6285 0.6136 0.59878 0.8367 0.8259 0.8148 0.8033 0.7916 0.7796 0.7673 0.7548 0.7420 0.72919 0.9090 0.9016 0.8939 0.8858 0.8774 0.8686 0.8596 0.8502 0.8405 0.8305
10 0.9531 0.9486 0.9437 0.9386 0.9332 0.9274 0.9214 0.9151 0.9084 0.901511 0.9776 0.9750 0.9723 0.9693 0.9661 0.9627 0.9591 0.9552 0.9510 0.946712 0.9900 0.9887 0.9873 0.9857 0.9840 0.9821 0.9801 0.9779 0.9755 0.973013 0.9958 0.9952 0.9945 0.9937 0.9929 0.9920 0.9909 0.9898 0.9885 0.987214 0.9984 0.9981 0.9978 0.9974 0.9970 0.9966 0.9961 0.9956 0.9950 0.9943
15 0.9994 0.9993 0.9992 0.9990 0.9988 0.9986 0.9984 0.9982 0.9979 0.997616 0.9998 0.9997 0.9997 0.9996 0.9996 0.9995 0.9994 0.9993 0.9992 0.999017 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997 0.9997 0.999618 1 1 1 1 0.9999 0.9999 0.9999 0.9999 0.9999 0.999919 1 1 1 1 1 1 1 1 1 1
11
P(X ≤ x) ou X ∼ P(λ)
λx 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.00 0.0008 0.0007 0.0007 0.0006 0.0006 0.0005 0.0005 0.0004 0.0004 0.00031 0.0067 0.0061 0.0056 0.0051 0.0047 0.0043 0.0039 0.0036 0.0033 0.00302 0.0275 0.0255 0.0236 0.0219 0.0203 0.0188 0.0174 0.0161 0.0149 0.01383 0.0767 0.0719 0.0674 0.0632 0.0591 0.0554 0.0518 0.0485 0.0453 0.04244 0.1641 0.1555 0.1473 0.1395 0.1321 0.1249 0.1181 0.1117 0.1055 0.0996
5 0.2881 0.2759 0.2640 0.2526 0.2414 0.2307 0.2203 0.2103 0.2006 0.19126 0.4349 0.4204 0.4060 0.3920 0.3782 0.3646 0.3514 0.3384 0.3257 0.31347 0.5838 0.5689 0.5541 0.5393 0.5246 0.5100 0.4956 0.4812 0.4670 0.45308 0.7160 0.7027 0.6892 0.6757 0.6620 0.6482 0.6343 0.6204 0.6065 0.59259 0.8202 0.8096 0.7988 0.7877 0.7764 0.7649 0.7531 0.7411 0.7290 0.7166
10 0.8942 0.8867 0.8788 0.8707 0.8622 0.8535 0.8445 0.8352 0.8257 0.815911 0.9420 0.9371 0.9319 0.9265 0.9208 0.9148 0.9085 0.9020 0.8952 0.888112 0.9703 0.9673 0.9642 0.9609 0.9573 0.9536 0.9496 0.9454 0.9409 0.936213 0.9857 0.9841 0.9824 0.9805 0.9784 0.9762 0.9739 0.9714 0.9687 0.965814 0.9935 0.9927 0.9918 0.9908 0.9897 0.9886 0.9873 0.9859 0.9844 0.9827
15 0.9972 0.9969 0.9964 0.9959 0.9954 0.9948 0.9941 0.9934 0.9926 0.991816 0.9989 0.9987 0.9985 0.9983 0.9980 0.9978 0.9974 0.9971 0.9967 0.996317 0.9996 0.9995 0.9994 0.9993 0.9992 0.9991 0.9989 0.9988 0.9986 0.998418 0.9998 0.9998 0.9998 0.9997 0.9997 0.9996 0.9996 0.9995 0.9994 0.999319 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997
20 1 1 1 1 1 1 0.9999 0.9999 0.9999 0.999921 1 1 1 1 1 1 1 1 1 1
P(X ≤ x) ou X ∼ P(λ)
λx 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.00 0.0003 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0001 0.00011 0.0028 0.0025 0.0023 0.0021 0.0019 0.0018 0.0016 0.0015 0.0014 0.00122 0.0127 0.0118 0.0109 0.0100 0.0093 0.0086 0.0079 0.0073 0.0068 0.00623 0.0396 0.0370 0.0346 0.0323 0.0301 0.0281 0.0262 0.0244 0.0228 0.02124 0.0940 0.0887 0.0837 0.0789 0.0744 0.0701 0.0660 0.0621 0.0584 0.0550
5 0.1822 0.1736 0.1653 0.1573 0.1496 0.1422 0.1352 0.1284 0.1219 0.11576 0.3013 0.2896 0.2781 0.2670 0.2562 0.2457 0.2355 0.2256 0.2160 0.20687 0.4391 0.4254 0.4119 0.3987 0.3856 0.3728 0.3602 0.3478 0.3357 0.32398 0.5786 0.5647 0.5507 0.5369 0.5231 0.5094 0.4958 0.4823 0.4689 0.45579 0.7041 0.6915 0.6788 0.6659 0.6530 0.6400 0.6269 0.6137 0.6006 0.5874
10 0.8058 0.7955 0.7850 0.7743 0.7634 0.7522 0.7409 0.7294 0.7178 0.706011 0.8807 0.8731 0.8652 0.8571 0.8487 0.8400 0.8311 0.8220 0.8126 0.803012 0.9313 0.9261 0.9207 0.9150 0.9091 0.9029 0.8965 0.8898 0.8829 0.875813 0.9628 0.9595 0.9561 0.9524 0.9486 0.9445 0.9403 0.9358 0.9311 0.926114 0.9810 0.9791 0.9771 0.9749 0.9726 0.9701 0.9675 0.9647 0.9617 0.9585
15 0.9908 0.9898 0.9887 0.9875 0.9862 0.9848 0.9832 0.9816 0.9798 0.978016 0.9958 0.9953 0.9947 0.9941 0.9934 0.9926 0.9918 0.9909 0.9899 0.988917 0.9982 0.9979 0.9977 0.9973 0.9970 0.9966 0.9962 0.9957 0.9952 0.994718 0.9992 0.9991 0.9990 0.9989 0.9987 0.9985 0.9983 0.9981 0.9978 0.997619 0.9997 0.9997 0.9996 0.9995 0.9995 0.9994 0.9993 0.9992 0.9991 0.9989
20 0.9999 0.9999 0.9998 0.9998 0.9998 0.9998 0.9997 0.9997 0.9996 0.999621 1 1 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.999822 1 1 1 1 1 1 1 1 0.9999 0.999923 1 1 1 1 1 1 1 1 1 1
12
P(X ≤ x) ou X ∼ P(λ)
λx 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.00 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.00001 0.0011 0.0010 0.0009 0.0009 0.0008 0.0007 0.0007 0.0006 0.0005 0.00052 0.0058 0.0053 0.0049 0.0045 0.0042 0.0038 0.0035 0.0033 0.0030 0.00283 0.0198 0.0184 0.0172 0.0160 0.0149 0.0138 0.0129 0.0120 0.0111 0.01034 0.0517 0.0486 0.0456 0.0429 0.0403 0.0378 0.0355 0.0333 0.0312 0.0293
5 0.1098 0.1041 0.0986 0.0935 0.0885 0.0838 0.0793 0.0750 0.0710 0.06716 0.1978 0.1892 0.1808 0.1727 0.1649 0.1574 0.1502 0.1433 0.1366 0.13017 0.3123 0.3010 0.2900 0.2792 0.2687 0.2584 0.2485 0.2388 0.2294 0.22028 0.4426 0.4296 0.4168 0.4042 0.3918 0.3796 0.3676 0.3558 0.3442 0.33289 0.5742 0.5611 0.5479 0.5349 0.5218 0.5089 0.4960 0.4832 0.4705 0.4579
10 0.6941 0.6820 0.6699 0.6576 0.6453 0.6329 0.6205 0.6080 0.5955 0.583011 0.7932 0.7832 0.7730 0.7626 0.7520 0.7412 0.7303 0.7193 0.7081 0.696812 0.8684 0.8607 0.8529 0.8448 0.8364 0.8279 0.8191 0.8101 0.8009 0.791613 0.9210 0.9156 0.9100 0.9042 0.8981 0.8919 0.8853 0.8786 0.8716 0.864514 0.9552 0.9517 0.9480 0.9441 0.9400 0.9357 0.9312 0.9265 0.9216 0.9165
15 0.9760 0.9738 0.9715 0.9691 0.9665 0.9638 0.9609 0.9579 0.9546 0.951316 0.9878 0.9865 0.9852 0.9838 0.9823 0.9806 0.9789 0.9770 0.9751 0.973017 0.9941 0.9934 0.9927 0.9919 0.9911 0.9902 0.9892 0.9881 0.9870 0.985718 0.9973 0.9969 0.9966 0.9962 0.9957 0.9952 0.9947 0.9941 0.9935 0.992819 0.9988 0.9986 0.9985 0.9983 0.9980 0.9978 0.9975 0.9972 0.9969 0.9965
20 0.9995 0.9994 0.9993 0.9992 0.9991 0.9990 0.9989 0.9987 0.9986 0.998421 0.9998 0.9998 0.9997 0.9997 0.9996 0.9996 0.9995 0.9995 0.9994 0.999322 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997 0.999723 1 1 1 1 0.9999 0.9999 0.9999 0.9999 0.9999 0.999924 1 1 1 1 1 1 1 1 1 1
P(X ≤ x) ou X ∼ P(λ)
λx 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.00 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00001 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00002 0.0012 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.00003 0.0049 0.0023 0.0011 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.00004 0.0151 0.0076 0.0037 0.0018 0.0009 0.0004 0.0002 0.0001 0.0000 0.0000
5 0.0375 0.0203 0.0107 0.0055 0.0028 0.0014 0.0007 0.0003 0.0002 0.00016 0.0786 0.0458 0.0259 0.0142 0.0076 0.0040 0.0021 0.0010 0.0005 0.00037 0.1432 0.0895 0.0540 0.0316 0.0180 0.0100 0.0054 0.0029 0.0015 0.00088 0.2320 0.1550 0.0998 0.0621 0.0374 0.0220 0.0126 0.0071 0.0039 0.00219 0.3405 0.2424 0.1658 0.1094 0.0699 0.0433 0.0261 0.0154 0.0089 0.0050
10 0.4599 0.3472 0.2517 0.1757 0.1185 0.0774 0.0491 0.0304 0.0183 0.010811 0.5793 0.4616 0.3532 0.2600 0.1848 0.1270 0.0847 0.0549 0.0347 0.021412 0.6887 0.5760 0.4631 0.3585 0.2676 0.1931 0.1350 0.0917 0.0606 0.039013 0.7813 0.6815 0.5730 0.4644 0.3632 0.2745 0.2009 0.1426 0.0984 0.066114 0.8540 0.7720 0.6751 0.5704 0.4657 0.3675 0.2808 0.2081 0.1497 0.1049
15 0.9074 0.8444 0.7636 0.6694 0.5681 0.4667 0.3715 0.2867 0.2148 0.156516 0.9441 0.8987 0.8355 0.7559 0.6641 0.5660 0.4677 0.3751 0.2920 0.221117 0.9678 0.9370 0.8905 0.8272 0.7489 0.6593 0.5640 0.4686 0.3784 0.297018 0.9823 0.9626 0.9302 0.8826 0.8195 0.7423 0.6550 0.5622 0.4695 0.381419 0.9907 0.9787 0.9573 0.9235 0.8752 0.8122 0.7363 0.6509 0.5606 0.4703
20 0.9953 0.9884 0.9750 0.9521 0.9170 0.8682 0.8055 0.7307 0.6472 0.559121 0.9977 0.9939 0.9859 0.9712 0.9469 0.9108 0.8615 0.7991 0.7255 0.643722 0.9990 0.9970 0.9924 0.9833 0.9673 0.9418 0.9047 0.8551 0.7931 0.720623 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.8989 0.8490 0.787524 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.9317 0.8933 0.8432
25 0.9999 0.9997 0.9990 0.9974 0.9938 0.9869 0.9748 0.9554 0.9269 0.887826 1 0.9999 0.9995 0.9987 0.9967 0.9925 0.9848 0.9718 0.9514 0.922127 1 0.9999 0.9998 0.9994 0.9983 0.9959 0.9912 0.9827 0.9687 0.947528 1 1 0.9999 0.9997 0.9991 0.9978 0.9950 0.9897 0.9805 0.965729 1 1 1 0.9999 0.9996 0.9989 0.9973 0.9941 0.9882 0.9782
13
1.3 Fonction de repartition de la loi Normale centree reduite
– Si X ∼ N (µ, σ2), alors f(x) = 1√2πσ2 exp
(− 12 (x−µ
σ )2), E(X) = µ et Var(X) = σ2.
– On note quelquefois U la v. a. gaussienne centree-reduite et Φ sa fonction de repartition :U ∼ N (0, 1).
– La table qui suit donne les valeurs de la fonction de repartition empirique de la loi normalecentree reduite Φ(x) pour les valeurs de x positives.
– Pour les valeurs negatives de x, on utilisera la relation Φ(x) = 1 − Φ(−x).
Φ(x) = P(X ≤ x) ou X ∼ N (0, 1) et x = x1 + x2
x2x1 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.00 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.10 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.20 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.30 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.40 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.50 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.60 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.70 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.80 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.90 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.00 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.86211.10 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.20 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.30 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.40 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.50 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94411.60 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.70 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.80 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.90 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.00 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.98172.10 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.20 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98902.30 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.40 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.50 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99522.60 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642.70 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99742.80 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99812.90 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.00 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.99903.10 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.99933.20 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.99953.30 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.99973.40 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
3.50 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.99983.60 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.99993.70 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.99993.80 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.99993.90 1 1 1 1 1 1 1 1 1 1
14
1.4 Fractiles de la loi Normale centree reduite
Pour les valeurs de α < 0.5, on utilisera la relation uα = −u1−α
uα = Φ−1(α) ou α = α1 + α2
α2α1 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
0.500 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0201 0.02260.510 0.0251 0.0276 0.0301 0.0326 0.0351 0.0376 0.0401 0.0426 0.0451 0.04760.520 0.0502 0.0527 0.0552 0.0577 0.0602 0.0627 0.0652 0.0677 0.0702 0.07280.530 0.0753 0.0778 0.0803 0.0828 0.0853 0.0878 0.0904 0.0929 0.0954 0.09790.540 0.1004 0.1030 0.1055 0.1080 0.1105 0.1130 0.1156 0.1181 0.1206 0.1231
0.550 0.1257 0.1282 0.1307 0.1332 0.1358 0.1383 0.1408 0.1434 0.1459 0.14840.560 0.1510 0.1535 0.1560 0.1586 0.1611 0.1637 0.1662 0.1687 0.1713 0.17380.570 0.1764 0.1789 0.1815 0.1840 0.1866 0.1891 0.1917 0.1942 0.1968 0.19930.580 0.2019 0.2045 0.2070 0.2096 0.2121 0.2147 0.2173 0.2198 0.2224 0.22500.590 0.2275 0.2301 0.2327 0.2353 0.2378 0.2404 0.2430 0.2456 0.2482 0.2508
0.600 0.2533 0.2559 0.2585 0.2611 0.2637 0.2663 0.2689 0.2715 0.2741 0.27670.610 0.2793 0.2819 0.2845 0.2871 0.2898 0.2924 0.2950 0.2976 0.3002 0.30290.620 0.3055 0.3081 0.3107 0.3134 0.3160 0.3186 0.3213 0.3239 0.3266 0.32920.630 0.3319 0.3345 0.3372 0.3398 0.3425 0.3451 0.3478 0.3505 0.3531 0.35580.640 0.3585 0.3611 0.3638 0.3665 0.3692 0.3719 0.3745 0.3772 0.3799 0.3826
0.650 0.3853 0.3880 0.3907 0.3934 0.3961 0.3989 0.4016 0.4043 0.4070 0.40970.660 0.4125 0.4152 0.4179 0.4207 0.4234 0.4261 0.4289 0.4316 0.4344 0.43720.670 0.4399 0.4427 0.4454 0.4482 0.4510 0.4538 0.4565 0.4593 0.4621 0.46490.680 0.4677 0.4705 0.4733 0.4761 0.4789 0.4817 0.4845 0.4874 0.4902 0.49300.690 0.4959 0.4987 0.5015 0.5044 0.5072 0.5101 0.5129 0.5158 0.5187 0.5215
0.700 0.5244 0.5273 0.5302 0.5330 0.5359 0.5388 0.5417 0.5446 0.5476 0.55050.710 0.5534 0.5563 0.5592 0.5622 0.5651 0.5681 0.5710 0.5740 0.5769 0.57990.720 0.5828 0.5858 0.5888 0.5918 0.5948 0.5978 0.6008 0.6038 0.6068 0.60980.730 0.6128 0.6158 0.6189 0.6219 0.6250 0.6280 0.6311 0.6341 0.6372 0.64030.740 0.6433 0.6464 0.6495 0.6526 0.6557 0.6588 0.6620 0.6651 0.6682 0.6713
0.750 0.6745 0.6776 0.6808 0.6840 0.6871 0.6903 0.6935 0.6967 0.6999 0.70310.760 0.7063 0.7095 0.7128 0.7160 0.7192 0.7225 0.7257 0.7290 0.7323 0.73560.770 0.7388 0.7421 0.7454 0.7488 0.7521 0.7554 0.7588 0.7621 0.7655 0.76880.780 0.7722 0.7756 0.7790 0.7824 0.7858 0.7892 0.7926 0.7961 0.7995 0.80300.790 0.8064 0.8099 0.8134 0.8169 0.8204 0.8239 0.8274 0.8310 0.8345 0.8381
0.800 0.8416 0.8452 0.8488 0.8524 0.8560 0.8596 0.8633 0.8669 0.8705 0.87420.810 0.8779 0.8816 0.8853 0.8890 0.8927 0.8965 0.9002 0.9040 0.9078 0.91160.820 0.9154 0.9192 0.9230 0.9269 0.9307 0.9346 0.9385 0.9424 0.9463 0.95020.830 0.9542 0.9581 0.9621 0.9661 0.9701 0.9741 0.9782 0.9822 0.9863 0.99040.840 0.9945 0.9986 1.0027 1.0069 1.0110 1.0152 1.0194 1.0237 1.0279 1.0322
0.850 1.0364 1.0407 1.0450 1.0494 1.0537 1.0581 1.0625 1.0669 1.0714 1.07580.860 1.0803 1.0848 1.0893 1.0939 1.0985 1.1031 1.1077 1.1123 1.1170 1.12170.870 1.1264 1.1311 1.1359 1.1407 1.1455 1.1503 1.1552 1.1601 1.1650 1.17000.880 1.1750 1.1800 1.1850 1.1901 1.1952 1.2004 1.2055 1.2107 1.2160 1.22120.890 1.2265 1.2319 1.2372 1.2426 1.2481 1.2536 1.2591 1.2646 1.2702 1.2759
0.900 1.2816 1.2873 1.2930 1.2988 1.3047 1.3106 1.3165 1.3225 1.3285 1.33460.910 1.3408 1.3469 1.3532 1.3595 1.3658 1.3722 1.3787 1.3852 1.3917 1.39840.920 1.4051 1.4118 1.4187 1.4255 1.4325 1.4395 1.4466 1.4538 1.4611 1.46840.930 1.4758 1.4833 1.4909 1.4985 1.5063 1.5141 1.5220 1.5301 1.5382 1.54640.940 1.5548 1.5632 1.5718 1.5805 1.5893 1.5982 1.6072 1.6164 1.6258 1.6352
0.950 1.6449 1.6546 1.6646 1.6747 1.6849 1.6954 1.7060 1.7169 1.7279 1.73920.960 1.7507 1.7624 1.7744 1.7866 1.7991 1.8119 1.8250 1.8384 1.8522 1.86630.970 1.8808 1.8957 1.9110 1.9268 1.9431 1.9600 1.9774 1.9954 2.0141 2.03350.980 2.0537 2.0749 2.0969 2.1201 2.1444 2.1701 2.1973 2.2262 2.2571 2.29040.990 2.3263 2.3656 2.4089 2.4573 2.5121 2.5758 2.6521 2.7478 2.8782 3.0902
15
1.5 Fractiles de la loi du χ2
Si X ∼ χ2ν , E(X) = ν et Var(X) = 2ν. Pour les valeurs de ν > 50, on utilisera la relation
χ2ν,α = (uα +
√2ν − 1)2/2.
χ2ν,α
αν 0.005 0.010 0.025 0.050 0.100 0.250 0.500 0.750 0.900 0.950 0.975 0.990 0.9951 0.0000393 0.000157 0.000982 0.00393 0.0158 0.102 0.455 1.32 2.71 3.84 5.02 6.63 7.882 0.0100 0.0201 0.0506 0.103 0.211 0.575 1.39 2.77 4.61 5.99 7.38 9.21 10.63 0.0717 0.115 0.216 0.352 0.584 1.21 2.37 4.11 6.25 7.81 9.35 11.3 12.84 0.207 0.297 0.484 0.711 1.06 1.92 3.36 5.39 7.78 9.49 11.1 13.3 14.95 0.412 0.554 0.831 1.15 1.61 2.67 4.35 6.63 9.24 11.1 12.8 15.1 16.76 0.676 0.872 1.24 1.64 2.20 3.45 5.35 7.84 10.6 12.6 14.4 16.8 18.57 0.989 1.24 1.69 2.17 2.83 4.25 6.35 9.04 12.0 14.1 16.0 18.5 20.38 1.34 1.65 2.18 2.73 3.49 5.07 7.34 10.2 13.4 15.5 17.5 20.1 22.09 1.73 2.09 2.70 3.33 4.17 5.90 8.34 11.4 14.7 16.9 19.0 21.7 23.610 2.16 2.56 3.25 3.94 4.87 6.74 9.34 12.5 16.0 18.3 20.5 23.2 25.211 2.60 3.05 3.82 4.57 5.58 7.58 10.3 13.7 17.3 19.7 21.9 24.7 26.812 3.07 3.57 4.40 5.23 6.30 8.44 11.3 14.8 18.5 21.0 23.3 26.2 28.313 3.57 4.11 5.01 5.89 7.04 9.30 12.3 16.0 19.8 22.4 24.7 27.7 29.814 4.07 4.66 5.63 6.57 7.79 10.2 13.3 17.1 21.1 23.7 26.1 29.1 31.315 4.60 5.23 6.26 7.26 8.55 11.0 14.3 18.2 22.3 25.0 27.5 30.6 32.816 5.14 5.81 6.91 7.96 9.31 11.9 15.3 19.4 23.5 26.3 28.8 32.0 34.317 5.70 6.41 7.56 8.67 10.1 12.8 16.3 20.5 24.8 27.6 30.2 33.4 35.718 6.26 7.01 8.23 9.39 10.9 13.7 17.3 21.6 26.0 28.9 31.5 34.8 37.219 6.84 7.63 8.91 10.1 11.7 14.6 18.3 22.7 27.2 30.1 32.9 36.2 38.620 7.43 8.26 9.59 10.9 12.4 15.5 19.3 23.8 28.4 31.4 34.2 37.6 40.021 8.03 8.90 10.3 11.6 13.2 16.3 20.3 24.9 29.6 32.7 35.5 38.9 41.422 8.64 9.54 11.0 12.3 14.0 17.2 21.3 26.0 30.8 33.9 36.8 40.3 42.823 9.26 10.2 11.7 13.1 14.8 18.1 22.3 27.1 32.0 35.2 38.1 41.6 44.224 9.89 10.9 12.4 13.8 15.7 19.0 23.3 28.2 33.2 36.4 39.4 43.0 45.625 10.5 11.5 13.1 14.6 16.5 19.9 24.3 29.3 34.4 37.7 40.6 44.3 46.926 11.2 12.2 13.8 15.4 17.3 20.8 25.3 30.4 35.6 38.9 41.9 45.6 48.327 11.8 12.9 14.6 16.2 18.1 21.7 26.3 31.5 36.7 40.1 43.2 47.0 49.628 12.5 13.6 15.3 16.9 18.9 22.7 27.3 32.6 37.9 41.3 44.5 48.3 51.029 13.1 14.3 16.0 17.7 19.8 23.6 28.3 33.7 39.1 42.6 45.7 49.6 52.330 13.8 15.0 16.8 18.5 20.6 24.5 29.3 34.8 40.3 43.8 47.0 50.9 53.731 14.5 15.7 17.5 19.3 21.4 25.4 30.3 35.9 41.4 45.0 48.2 52.2 55.032 15.1 16.4 18.3 20.1 22.3 26.3 31.3 37.0 42.6 46.2 49.5 53.5 56.333 15.8 17.1 19.0 20.9 23.1 27.2 32.3 38.1 43.7 47.4 50.7 54.8 57.634 16.5 17.8 19.8 21.7 24.0 28.1 33.3 39.1 44.9 48.6 52.0 56.1 59.035 17.2 18.5 20.6 22.5 24.8 29.1 34.3 40.2 46.1 49.8 53.2 57.3 60.336 17.9 19.2 21.3 23.3 25.6 30.0 35.3 41.3 47.2 51.0 54.4 58.6 61.637 18.6 20.0 22.1 24.1 26.5 30.9 36.3 42.4 48.4 52.2 55.7 59.9 62.938 19.3 20.7 22.9 24.9 27.3 31.8 37.3 43.5 49.5 53.4 56.9 61.2 64.239 20.0 21.4 23.7 25.7 28.2 32.7 38.3 44.5 50.7 54.6 58.1 62.4 65.540 20.7 22.2 24.4 26.5 29.1 33.7 39.3 45.6 51.8 55.8 59.3 63.7 66.841 21.4 22.9 25.2 27.3 29.9 34.6 40.3 46.7 52.9 56.9 60.6 65.0 68.142 22.1 23.7 26.0 28.1 30.8 35.5 41.3 47.8 54.1 58.1 61.8 66.2 69.343 22.9 24.4 26.8 29.0 31.6 36.4 42.3 48.8 55.2 59.3 63.0 67.5 70.644 23.6 25.1 27.6 29.8 32.5 37.4 43.3 49.9 56.4 60.5 64.2 68.7 71.945 24.3 25.9 28.4 30.6 33.4 38.3 44.3 51.0 57.5 61.7 65.4 70.0 73.246 25.0 26.7 29.2 31.4 34.2 39.2 45.3 52.1 58.6 62.8 66.6 71.2 74.447 25.8 27.4 30.0 32.3 35.1 40.1 46.3 53.1 59.8 64.0 67.8 72.4 75.748 26.5 28.2 30.8 33.1 35.9 41.1 47.3 54.2 60.9 65.2 69.0 73.7 77.049 27.2 28.9 31.6 33.9 36.8 42.0 48.3 55.3 62.0 66.3 70.2 74.9 78.250 28.0 29.7 32.4 34.8 37.7 42.9 49.3 56.3 63.2 67.5 71.4 76.2 79.5
16
1.6 Fractiles de la loi de Student
Pour les valeurs de α ≤ 0.5, on utilisera la relation tν,α = −tν,1−α.
tν,α
αν 0.6 0.75 0.9 0.95 0.975 0.99 0.995 0.99951 0.325 1.000 3.078 6.314 12.706 31.821 63.657 636.6192 0.289 0.816 1.886 2.920 4.303 6.965 9.925 31.5993 0.277 0.765 1.638 2.353 3.182 4.541 5.841 12.9244 0.271 0.741 1.533 2.132 2.776 3.747 4.604 8.6105 0.267 0.727 1.476 2.015 2.571 3.365 4.032 6.869
6 0.265 0.718 1.440 1.943 2.447 3.143 3.707 5.9597 0.263 0.711 1.415 1.895 2.365 2.998 3.499 5.4088 0.262 0.706 1.397 1.860 2.306 2.896 3.355 5.0419 0.261 0.703 1.383 1.833 2.262 2.821 3.250 4.78110 0.260 0.700 1.372 1.812 2.228 2.764 3.169 4.587
11 0.260 0.697 1.363 1.796 2.201 2.718 3.106 4.43712 0.259 0.695 1.356 1.782 2.179 2.681 3.055 4.31813 0.259 0.694 1.350 1.771 2.160 2.650 3.012 4.22114 0.258 0.692 1.345 1.761 2.145 2.624 2.977 4.14015 0.258 0.691 1.341 1.753 2.131 2.602 2.947 4.073
16 0.258 0.690 1.337 1.746 2.120 2.583 2.921 4.01517 0.257 0.689 1.333 1.740 2.110 2.567 2.898 3.96518 0.257 0.688 1.330 1.734 2.101 2.552 2.878 3.92219 0.257 0.688 1.328 1.729 2.093 2.539 2.861 3.88320 0.257 0.687 1.325 1.725 2.086 2.528 2.845 3.850
21 0.257 0.686 1.323 1.721 2.080 2.518 2.831 3.81922 0.256 0.686 1.321 1.717 2.074 2.508 2.819 3.79223 0.256 0.685 1.319 1.714 2.069 2.500 2.807 3.76824 0.256 0.685 1.318 1.711 2.064 2.492 2.797 3.74525 0.256 0.684 1.316 1.708 2.060 2.485 2.787 3.725
26 0.256 0.684 1.315 1.706 2.056 2.479 2.779 3.70727 0.256 0.684 1.314 1.703 2.052 2.473 2.771 3.69028 0.256 0.683 1.313 1.701 2.048 2.467 2.763 3.67429 0.256 0.683 1.311 1.699 2.045 2.462 2.756 3.65930 0.256 0.683 1.310 1.697 2.042 2.457 2.750 3.646
40 0.255 0.681 1.303 1.684 2.021 2.423 2.704 3.55160 0.254 0.679 1.296 1.671 2.000 2.390 2.660 3.460120 0.254 0.677 1.289 1.658 1.980 2.358 2.617 3.3731000 0.253 0.675 1.282 1.646 1.962 2.330 2.581 3.300
17
1.7
Fra
ctiles
de
lalo
ide
Fis
her
Pou
rle
spe
tite
sva
leur
sde
α≤
0.5,
onut
ilise
rala
rela
tion
:F
ν1,ν
2,α
=1/
Fν2,ν
1,1−
α.
Fν1,ν
2,0
.90
ν1
ν2
12
34
56
78
910
12
15
20
24
30
40
60
120
∞1
39.8
649.5
053.5
955.8
357.2
458.2
058.9
159.4
459.8
660.1
960.7
161.2
261.7
462.0
062.2
662.5
362.7
963.0
666.1
22
8.5
39.0
09.1
69.2
49.2
99.3
39.3
59.3
79.3
89.3
99.4
19.4
29.4
49.4
59.4
69.4
79.4
79.4
89.4
93
5.5
45.4
65.3
95.3
45.3
15.2
85.2
75.2
55.2
45.2
35.2
25.2
05.1
85.1
85.1
75.1
65.1
55.1
45.1
34
4.5
44.3
24.1
94.1
14.0
54.0
13.9
83.9
53.9
43.9
23.9
03.8
73.8
43.8
33.8
23.8
03.7
93.7
83.7
65
4.0
63.7
83.6
23.5
23.4
53.4
03.3
73.3
43.3
23.3
03.2
73.2
43.2
13.1
93.1
73.1
63.1
43.1
23.1
0
63.7
83.4
63.2
93.1
83.1
13.0
53.0
12.9
82.9
62.9
42.9
02.8
72.8
42.8
22.8
02.7
82.7
62.7
42.7
27
3.5
93.2
63.0
72.9
62.8
82.8
32.7
82.7
52.7
22.7
02.6
72.6
32.5
92.5
82.5
62.5
42.5
12.4
92.4
78
3.4
63.1
12.9
22.8
12.7
32.6
72.6
22.5
92.5
62.5
42.5
02.4
62.4
22.4
02.3
82.3
62.3
42.3
22.2
99
3.3
63.0
12.8
12.6
92.6
12.5
52.5
12.4
72.4
42.4
22.3
82.3
42.3
02.2
82.2
52.2
32.2
12.1
82.1
610
3.2
92.9
22.7
32.6
12.5
22.4
62.4
12.3
82.3
52.3
22.2
82.2
42.2
02.1
82.1
62.1
32.1
12.0
82.0
6
11
3.2
32.8
62.6
62.5
42.4
52.3
92.3
42.3
02.2
72.2
52.2
12.1
72.1
22.1
02.0
82.0
52.0
32.0
01.9
712
3.1
82.8
12.6
12.4
82.3
92.3
32.2
82.2
42.2
12.1
92.1
52.1
02.0
62.0
42.0
11.9
91.9
61.9
31.9
013
3.1
42.7
62.5
62.4
32.3
52.2
82.2
32.2
02.1
62.1
42.1
02.0
52.0
11.9
81.9
61.9
31.9
01.8
81.8
514
3.1
02.7
32.5
22.3
92.3
12.2
42.1
92.1
52.1
22.1
02.0
52.0
11.9
61.9
41.9
11.8
91.8
61.8
31.8
015
3.0
72.7
02.4
92.3
62.2
72.2
12.1
62.1
22.0
92.0
62.0
21.9
71.9
21.9
01.8
71.8
51.8
21.7
91.7
6
16
3.0
52.6
72.4
62.3
32.2
42.1
82.1
32.0
92.0
62.0
31.9
91.9
41.8
91.8
71.8
41.8
11.7
81.7
51.7
217
3.0
32.6
42.4
42.3
12.2
22.1
52.1
02.0
62.0
32.0
01.9
61.9
11.8
61.8
41.8
11.7
81.7
51.7
21.6
918
3.0
12.6
22.4
22.2
92.2
02.1
32.0
82.0
42.0
01.9
81.9
31.8
91.8
41.8
11.7
81.7
51.7
21.6
91.6
619
2.9
92.6
12.4
02.2
72.1
82.1
12.0
62.0
21.9
81.9
61.9
11.8
61.8
11.7
91.7
61.7
31.7
01.6
71.6
320
2.9
72.5
92.3
82.2
52.1
62.0
92.0
42.0
01.9
61.9
41.8
91.8
41.7
91.7
71.7
41.7
11.6
81.6
41.6
1
21
2.9
62.5
72.3
62.2
32.1
42.0
82.0
21.9
81.9
51.9
21.8
71.8
31.7
81.7
51.7
21.6
91.6
61.6
21.5
922
2.9
52.5
62.3
52.2
22.1
32.0
62.0
11.9
71.9
31.9
01.8
61.8
11.7
61.7
31.7
01.6
71.6
41.6
01.5
723
2.9
42.5
52.3
42.2
12.1
12.0
51.9
91.9
51.9
21.8
91.8
41.8
01.7
41.7
21.6
91.6
61.6
21.5
91.5
524
2.9
32.5
42.3
32.1
92.1
02.0
41.9
81.9
41.9
11.8
81.8
31.7
81.7
31.7
01.6
71.6
41.6
11.5
71.5
325
2.9
22.5
32.3
22.1
82.0
92.0
21.9
71.9
31.8
91.8
71.8
21.7
71.7
21.6
91.6
61.6
31.5
91.5
61.5
2
26
2.9
12.5
22.3
12.1
72.0
82.0
11.9
61.9
21.8
81.8
61.8
11.7
61.7
11.6
81.6
51.6
11.5
81.5
41.5
027
2.9
02.5
12.3
02.1
72.0
72.0
01.9
51.9
11.8
71.8
51.8
01.7
51.7
01.6
71.6
41.6
01.5
71.5
31.4
928
2.8
92.5
02.2
92.1
62.0
62.0
01.9
41.9
01.8
71.8
41.7
91.7
41.6
91.6
61.6
31.5
91.5
61.5
21.4
829
2.8
92.5
02.2
82.1
52.0
61.9
91.9
31.8
91.8
61.8
31.7
81.7
31.6
81.6
51.6
21.5
81.5
51.5
11.4
730
2.8
82.4
92.2
82.1
42.0
51.9
81.9
31.8
81.8
51.8
21.7
71.7
21.6
71.6
41.6
11.5
71.5
41.5
01.4
6
40
2.8
42.4
42.2
32.0
92.0
01.9
31.8
71.8
31.7
91.7
61.7
11.6
61.6
11.5
71.5
41.5
11.4
71.4
21.3
860
2.7
92.3
92.1
82.0
41.9
51.8
71.8
21.7
71.7
41.7
11.6
61.6
01.5
41.5
11.4
81.4
41.4
01.3
51.2
9120
2.7
52.3
52.1
31.9
91.9
01.8
21.7
71.7
21.6
81.6
51.6
01.5
51.4
81.4
51.4
11.3
71.3
21.2
61.1
9∞
2.7
12.3
02.0
81.9
41.8
51.7
71.7
21.6
71.6
31.6
01.5
51.4
91.4
21.3
81.3
41.3
01.2
41.1
71.0
0
18
Fν1,ν
2,0
.95
ν1
ν2
12
34
56
78
910
12
15
20
24
30
40
60
120
∞1
161.4
199.5
215.7
224.6
230.2
234.0
236.8
238.9
240.5
241.9
243.9
245.9
248.0
249.1
250.1
251.1
252.2
253.3
395.4
218.5
119.0
019.1
619.2
519.3
019.3
319.3
519.3
719.3
819.4
019.4
119.4
319.4
519.4
519.4
619.4
719.4
819.4
919.5
03
10.1
39.5
59.2
89.1
29.0
18.9
48.8
98.8
58.8
18.7
98.7
48.7
08.6
68.6
48.6
28.5
98.5
78.5
58.5
34
7.7
16.9
46.5
96.3
96.2
66.1
66.0
96.0
46.0
05.9
65.9
15.8
65.8
05.7
75.7
55.7
25.6
95.6
65.6
35
6.6
15.7
95.4
15.1
95.0
54.9
54.8
84.8
24.7
74.7
44.6
84.6
24.5
64.5
34.5
04.4
64.4
34.4
04.3
7
65.9
95.1
44.7
64.5
34.3
94.2
84.2
14.1
54.1
04.0
64.0
03.9
43.8
73.8
43.8
13.7
73.7
43.7
03.6
77
5.5
94.7
44.3
54.1
23.9
73.8
73.7
93.7
33.6
83.6
43.5
73.5
13.4
43.4
13.3
83.3
43.3
03.2
73.2
38
5.3
24.4
64.0
73.8
43.6
93.5
83.5
03.4
43.3
93.3
53.2
83.2
23.1
53.1
23.0
83.0
43.0
12.9
72.9
39
5.1
24.2
63.8
63.6
33.4
83.3
73.2
93.2
33.1
83.1
43.0
73.0
12.9
42.9
02.8
62.8
32.7
92.7
52.7
110
4.9
64.1
03.7
13.4
83.3
33.2
23.1
43.0
73.0
22.9
82.9
12.8
52.7
72.7
42.7
02.6
62.6
22.5
82.5
4
11
4.8
43.9
83.5
93.3
63.2
03.0
93.0
12.9
52.9
02.8
52.7
92.7
22.6
52.6
12.5
72.5
32.4
92.4
52.4
012
4.7
53.8
93.4
93.2
63.1
13.0
02.9
12.8
52.8
02.7
52.6
92.6
22.5
42.5
12.4
72.4
32.3
82.3
42.3
013
4.6
73.8
13.4
13.1
83.0
32.9
22.8
32.7
72.7
12.6
72.6
02.5
32.4
62.4
22.3
82.3
42.3
02.2
52.2
114
4.6
03.7
43.3
43.1
12.9
62.8
52.7
62.7
02.6
52.6
02.5
32.4
62.3
92.3
52.3
12.2
72.2
22.1
82.1
315
4.5
43.6
83.2
93.0
62.9
02.7
92.7
12.6
42.5
92.5
42.4
82.4
02.3
32.2
92.2
52.2
02.1
62.1
12.0
7
16
4.4
93.6
33.2
43.0
12.8
52.7
42.6
62.5
92.5
42.4
92.4
22.3
52.2
82.2
42.1
92.1
52.1
12.0
62.0
117
4.4
53.5
93.2
02.9
62.8
12.7
02.6
12.5
52.4
92.4
52.3
82.3
12.2
32.1
92.1
52.1
02.0
62.0
11.9
618
4.4
13.5
53.1
62.9
32.7
72.6
62.5
82.5
12.4
62.4
12.3
42.2
72.1
92.1
52.1
12.0
62.0
21.9
71.9
219
4.3
83.5
23.1
32.9
02.7
42.6
32.5
42.4
82.4
22.3
82.3
12.2
32.1
62.1
12.0
72.0
31.9
81.9
31.8
820
4.3
53.4
93.1
02.8
72.7
12.6
02.5
12.4
52.3
92.3
52.2
82.2
02.1
22.0
82.0
41.9
91.9
51.9
01.8
4
21
4.3
23.4
73.0
72.8
42.6
82.5
72.4
92.4
22.3
72.3
22.2
52.1
82.1
02.0
52.0
11.9
61.9
21.8
71.8
122
4.3
03.4
43.0
52.8
22.6
62.5
52.4
62.4
02.3
42.3
02.2
32.1
52.0
72.0
31.9
81.9
41.8
91.8
41.7
823
4.2
83.4
23.0
32.8
02.6
42.5
32.4
42.3
72.3
22.2
72.2
02.1
32.0
52.0
11.9
61.9
11.8
61.8
11.7
624
4.2
63.4
03.0
12.7
82.6
22.5
12.4
22.3
62.3
02.2
52.1
82.1
12.0
31.9
81.9
41.8
91.8
41.7
91.7
325
4.2
43.3
92.9
92.7
62.6
02.4
92.4
02.3
42.2
82.2
42.1
62.0
92.0
11.9
61.9
21.8
71.8
21.7
71.7
1
26
4.2
33.3
72.9
82.7
42.5
92.4
72.3
92.3
22.2
72.2
22.1
52.0
71.9
91.9
51.9
01.8
51.8
01.7
51.6
927
4.2
13.3
52.9
62.7
32.5
72.4
62.3
72.3
12.2
52.2
02.1
32.0
61.9
71.9
31.8
81.8
41.7
91.7
31.6
728
4.2
03.3
42.9
52.7
12.5
62.4
52.3
62.2
92.2
42.1
92.1
22.0
41.9
61.9
11.8
71.8
21.7
71.7
11.6
529
4.1
83.3
32.9
32.7
02.5
52.4
32.3
52.2
82.2
22.1
82.1
02.0
31.9
41.9
01.8
51.8
11.7
51.7
01.6
430
4.1
73.3
22.9
22.6
92.5
32.4
22.3
32.2
72.2
12.1
62.0
92.0
11.9
31.8
91.8
41.7
91.7
41.6
81.6
2
40
4.0
83.2
32.8
42.6
12.4
52.3
42.2
52.1
82.1
22.0
82.0
01.9
21.8
41.7
91.7
41.6
91.6
41.5
81.5
160
4.0
03.1
52.7
62.5
32.3
72.2
52.1
72.1
02.0
41.9
91.9
21.8
41.7
51.7
01.6
51.5
91.5
31.4
71.3
9120
3.9
23.0
72.6
82.4
52.2
92.1
82.0
92.0
21.9
61.9
11.8
31.7
51.6
61.6
11.5
51.5
01.4
31.3
51.2
5∞
3.8
43.0
02.6
02.3
72.2
12.1
02.0
11.9
41.8
81.8
31.7
51.6
71.5
71.5
21.4
61.3
91.3
21.2
21.0
0
19
Fν1,ν
2,0
.975
ν1
ν2
12
34
56
78
910
12
15
20
24
30
40
60
120
∞1
647.8
799.5
864.2
899.6
921.9
937.1
948.2
956.7
963.3
968.6
976.7
984.9
993.1
997.2
1001
1005
1009
1014
1018
238.5
139.0
039.1
739.2
539.3
039.3
339.3
639.3
739.3
939.4
039.4
139.4
339.4
539.4
639.4
639.4
739.4
839.4
939.5
03
17.4
416.0
415.4
415.1
014.8
814.7
314.6
214.5
414.4
714.4
214.3
414.2
514.1
714.1
214.0
814.0
413.9
913.9
513.9
04
12.2
210.6
59.9
89.6
09.3
69.2
09.0
78.9
88.9
08.8
48.7
58.6
68.5
68.5
18.4
68.4
18.3
68.3
18.2
65
10.0
18.4
37.7
67.3
97.1
56.9
86.8
56.7
66.6
86.6
26.5
26.4
36.3
36.2
86.2
36.1
86.1
26.0
76.0
2
68.8
17.2
66.6
06.2
35.9
95.8
25.7
05.6
05.5
25.4
65.3
75.2
75.1
75.1
25.0
75.0
14.9
64.9
04.8
57
8.0
76.5
45.8
95.5
25.2
95.1
24.9
94.9
04.8
24.7
64.6
74.5
74.4
74.4
14.3
64.3
14.2
54.2
04.1
48
7.5
76.0
65.4
25.0
54.8
24.6
54.5
34.4
34.3
64.3
04.2
04.1
04.0
03.9
53.8
93.8
43.7
83.7
33.6
79
7.2
15.7
15.0
84.7
24.4
84.3
24.2
04.1
04.0
33.9
63.8
73.7
73.6
73.6
13.5
63.5
13.4
53.3
93.3
310
6.9
45.4
64.8
34.4
74.2
44.0
73.9
53.8
53.7
83.7
23.6
23.5
23.4
23.3
73.3
13.2
63.2
03.1
43.0
8
11
6.7
25.2
64.6
34.2
84.0
43.8
83.7
63.6
63.5
93.5
33.4
33.3
33.2
33.1
73.1
23.0
63.0
02.9
42.8
812
6.5
55.1
04.4
74.1
23.8
93.7
33.6
13.5
13.4
43.3
73.2
83.1
83.0
73.0
22.9
62.9
12.8
52.7
92.7
213
6.4
14.9
74.3
54.0
03.7
73.6
03.4
83.3
93.3
13.2
53.1
53.0
52.9
52.8
92.8
42.7
82.7
22.6
62.6
014
6.3
04.8
64.2
43.8
93.6
63.5
03.3
83.2
93.2
13.1
53.0
52.9
52.8
42.7
92.7
32.6
72.6
12.5
52.4
915
6.2
04.7
74.1
53.8
03.5
83.4
13.2
93.2
03.1
23.0
62.9
62.8
62.7
62.7
02.6
42.5
92.5
22.4
62.4
0
16
6.1
24.6
94.0
83.7
33.5
03.3
43.2
23.1
23.0
52.9
92.8
92.7
92.6
82.6
32.5
72.5
12.4
52.3
82.3
217
6.0
44.6
24.0
13.6
63.4
43.2
83.1
63.0
62.9
82.9
22.8
22.7
22.6
22.5
62.5
02.4
42.3
82.3
22.2
518
5.9
84.5
63.9
53.6
13.3
83.2
23.1
03.0
12.9
32.8
72.7
72.6
72.5
62.5
02.4
42.3
82.3
22.2
62.1
919
5.9
24.5
13.9
03.5
63.3
33.1
73.0
52.9
62.8
82.8
22.7
22.6
22.5
12.4
52.3
92.3
32.2
72.2
02.1
320
5.8
74.4
63.8
63.5
13.2
93.1
33.0
12.9
12.8
42.7
72.6
82.5
72.4
62.4
12.3
52.2
92.2
22.1
62.0
9
21
5.8
34.4
23.8
23.4
83.2
53.0
92.9
72.8
72.8
02.7
32.6
42.5
32.4
22.3
72.3
12.2
52.1
82.1
12.0
422
5.7
94.3
83.7
83.4
43.2
23.0
52.9
32.8
42.7
62.7
02.6
02.5
02.3
92.3
32.2
72.2
12.1
42.0
82.0
023
5.7
54.3
53.7
53.4
13.1
83.0
22.9
02.8
12.7
32.6
72.5
72.4
72.3
62.3
02.2
42.1
82.1
12.0
41.9
724
5.7
24.3
23.7
23.3
83.1
52.9
92.8
72.7
82.7
02.6
42.5
42.4
42.3
32.2
72.2
12.1
52.0
82.0
11.9
425
5.6
94.2
93.6
93.3
53.1
32.9
72.8
52.7
52.6
82.6
12.5
12.4
12.3
02.2
42.1
82.1
22.0
51.9
81.9
1
26
5.6
64.2
73.6
73.3
33.1
02.9
42.8
22.7
32.6
52.5
92.4
92.3
92.2
82.2
22.1
62.0
92.0
31.9
51.8
827
5.6
34.2
43.6
53.3
13.0
82.9
22.8
02.7
12.6
32.5
72.4
72.3
62.2
52.1
92.1
32.0
72.0
01.9
31.8
528
5.6
14.2
23.6
33.2
93.0
62.9
02.7
82.6
92.6
12.5
52.4
52.3
42.2
32.1
72.1
12.0
51.9
81.9
11.8
329
5.5
94.2
03.6
13.2
73.0
42.8
82.7
62.6
72.5
92.5
32.4
32.3
22.2
12.1
52.0
92.0
31.9
61.8
91.8
130
5.5
74.1
83.5
93.2
53.0
32.8
72.7
52.6
52.5
72.5
12.4
12.3
12.2
02.1
42.0
72.0
11.9
41.8
71.7
9
40
5.4
24.0
53.4
63.1
32.9
02.7
42.6
22.5
32.4
52.3
92.2
92.1
82.0
72.0
11.9
41.8
81.8
01.7
21.6
460
5.2
93.9
33.3
43.0
12.7
92.6
32.5
12.4
12.3
32.2
72.1
72.0
61.9
41.8
81.8
21.7
41.6
71.5
81.4
8120
5.1
53.8
03.2
32.8
92.6
72.5
22.3
92.3
02.2
22.1
62.0
51.9
41.8
21.7
61.6
91.6
11.5
31.4
31.3
1∞
5.0
23.6
93.1
22.7
92.5
72.4
12.2
92.1
92.1
12.0
51.9
41.8
31.7
11.6
41.5
71.4
81.3
91.2
71.0
0
20
Fν1,ν
2,0
.99
ν1
ν2
12
34
56
78
910
12
15
20
24
30
40
60
120
∞1
4052
4999.5
5403
5625
5764
5859
5928
5982
6022
6056
6106
6157
6209
6235
6261
6287
6313
6339
6366
298.5
099.0
099.1
799.2
599.3
099.3
399.3
699.3
799.3
999.4
099.4
299.4
399.4
599.4
699.4
799.4
799.4
899.4
999.5
93
34.1
230.8
229.4
628.7
128.2
427.9
127.6
727.4
927.3
527.2
327.0
526.8
726.6
926.6
026.5
026.4
126.3
226.2
226.1
24
21.2
018.0
016.6
915.9
815.5
215.2
114.9
814.8
014.6
614.5
514.3
714.2
014.0
213.9
313.8
413.7
513.6
513.5
613.4
65
16.2
613.2
712.0
611.3
910.9
710.6
710.4
610.2
910.1
610.0
59.8
99.7
29.5
59.4
79.3
89.2
99.2
09.1
19.0
2
613.7
510.9
29.7
89.1
58.7
58.4
78.2
68.1
07.9
87.8
77.7
27.5
67.4
07.3
17.2
37.1
47.0
66.9
76.8
87
12.2
59.5
58.4
57.8
57.4
67.1
96.9
96.8
46.7
26.6
26.4
76.3
16.1
66.0
75.9
95.9
15.8
25.7
45.6
58
11.2
68.6
57.5
97.0
16.6
36.3
76.1
86.0
35.9
15.8
15.6
75.5
25.3
65.2
85.2
05.1
25.0
34.9
54.8
69
10.5
68.0
26.9
96.4
26.0
65.8
05.6
15.4
75.3
55.2
65.1
14.9
64.8
14.7
34.6
54.5
74.4
84.4
04.3
110
10.0
47.5
66.5
55.9
95.6
45.3
95.2
05.0
64.9
44.8
54.7
14.5
64.4
14.3
34.2
54.1
74.0
84.0
03.9
1
11
9.6
57.2
16.2
25.6
75.3
25.0
74.8
94.7
44.6
34.5
44.4
04.2
54.1
04.0
23.9
43.8
63.7
83.6
93.6
012
9.3
36.9
35.9
55.4
15.0
64.8
24.6
44.5
04.3
94.3
04.1
64.0
13.8
63.7
83.7
03.6
23.5
43.4
53.3
613
9.0
76.7
05.7
45.2
14.8
64.6
24.4
44.3
04.1
94.1
03.9
63.8
23.6
63.5
93.5
13.4
33.3
43.2
53.1
714
8.8
66.5
15.5
65.0
44.6
94.4
64.2
84.1
44.0
33.9
43.8
03.6
63.5
13.4
33.3
53.2
73.1
83.0
93.0
015
8.6
86.3
65.4
24.8
94.5
64.3
24.1
44.0
03.8
93.8
03.6
73.5
23.3
73.2
93.2
13.1
33.0
52.9
62.8
7
16
8.5
36.2
35.2
94.7
74.4
44.2
04.0
33.8
93.7
83.6
93.5
53.4
13.2
63.1
83.1
03.0
22.9
32.8
42.7
517
8.4
06.1
15.1
84.6
74.3
44.1
03.9
33.7
93.6
83.5
93.4
63.3
13.1
63.0
83.0
02.9
22.8
32.7
52.6
518
8.2
96.0
15.0
94.5
84.2
54.0
13.8
43.7
13.6
03.5
13.3
73.2
33.0
83.0
02.9
22.8
42.7
52.6
62.5
719
8.1
85.9
35.0
14.5
04.1
73.9
43.7
73.6
33.5
23.4
33.3
03.1
53.0
02.9
22.8
42.7
62.6
72.5
82.4
920
8.1
05.8
54.9
44.4
34.1
03.8
73.7
03.5
63.4
63.3
73.2
33.0
92.9
42.8
62.7
82.6
92.6
12.5
22.4
2
21
8.0
25.7
84.8
74.3
74.0
43.8
13.6
43.5
13.4
03.3
13.1
73.0
32.8
82.8
02.7
22.6
42.5
52.4
62.3
622
7.9
55.7
24.8
24.3
13.9
93.7
63.5
93.4
53.3
53.2
63.1
22.9
82.8
32.7
52.6
72.5
82.5
02.4
02.3
123
7.8
85.6
64.7
64.2
63.9
43.7
13.5
43.4
13.3
03.2
13.0
72.9
32.7
82.7
02.6
22.5
42.4
52.3
52.2
624
7.8
25.6
14.7
24.2
23.9
03.6
73.5
03.3
63.2
63.1
73.0
32.8
92.7
42.6
62.5
82.4
92.4
02.3
12.2
125
7.7
75.5
74.6
84.1
83.8
53.6
33.4
63.3
23.2
23.1
32.9
92.8
52.7
02.6
22.5
42.4
52.3
62.2
72.1
7
26
7.7
25.5
34.6
44.1
43.8
23.5
93.4
23.2
93.1
83.0
92.9
62.8
12.6
62.5
82.5
02.4
22.3
32.2
32.1
327
7.6
85.4
94.6
04.1
13.7
83.5
63.3
93.2
63.1
53.0
62.9
32.7
82.6
32.5
52.4
72.3
82.2
92.2
02.1
028
7.6
45.4
54.5
74.0
73.7
53.5
33.3
63.2
33.1
23.0
32.9
02.7
52.6
02.5
22.4
42.3
52.2
62.1
72.0
629
7.6
05.4
24.5
44.0
43.7
33.5
03.3
33.2
03.0
93.0
02.8
72.7
32.5
72.4
92.4
12.3
32.2
32.1
42.0
330
7.5
65.3
94.5
14.0
23.7
03.4
73.3
03.1
73.0
72.9
82.8
42.7
02.5
52.4
72.3
92.3
02.2
12.1
12.0
1
40
7.3
15.1
84.3
13.8
33.5
13.2
93.1
22.9
92.8
92.8
02.6
62.5
22.3
72.2
92.2
02.1
12.0
21.9
21.8
060
7.0
84.9
84.1
33.6
53.3
43.1
22.9
52.8
22.7
22.6
32.5
02.3
52.2
02.1
22.0
31.9
41.8
41.7
31.6
0120
6.8
54.7
93.9
53.4
83.1
72.9
62.7
92.6
62.5
62.4
72.3
42.1
92.0
31.9
51.8
61.7
61.6
61.5
31.3
8∞
6.6
34.6
13.7
83.3
23.0
22.8
02.6
42.5
12.4
12.3
22.1
82.0
41.8
81.7
91.7
01.5
91.4
71.3
21.0
0
21
2 Intervalles de confiance pour une proportion
2.1 Intervalle bilateral (1− α = 0.90) et intervalle unilateral (1− α = 0.95)
22
2.2 Intervalle bilateral (1−α = 0.95) et intervalle unilateral (1−α = 0.975)
23
2.3 Intervalle bilateral (1− α = 0.98) et intervalle unilateral (1− α = 0.99)
24
2.4 Intervalle bilateral (1−α = 0.99) et intervalle unilateral (1−α = 0.995)
25
3 Puissance du test de Student
3.1 Tests bilateraux pour α = 0.05
26
3.2 Tests bilateraux pour α = 0.01
27
3.3 Tests unilateraux pour α = 0.05
28
3.4 Tests unilateraux pour α = 0.01
29
4 Test de Wilcoxon
Soient X1, . . . , Xn1 et Y1, . . . , Yn2 les deux echantillons. Par convention on suppose n1 ≤ n2. Onnote WX la somme des rangs des observations issues de l’echantillon de X .
4.1 Test bilateral
On rejette H0 : FX = FY par rapport a H1 : FX = FY si WX ≤ B ou WX ≥ n1(n1 +n2 +1)−B,B etant la valeur lue dans la table.
α = 5%
n1 2 3 4 5 6 7 8 9 10 11 12 13 14 15n24 105 6 11 176 7 12 18 267 7 13 20 27 368 3 8 14 21 29 38 499 3 8 15 22 31 40 51 6310 3 9 15 23 32 42 53 65 7811 4 9 16 24 34 44 55 68 81 9612 4 10 17 26 35 46 58 71 85 99 11513 4 10 18 27 37 48 60 73 88 103 119 13714 4 11 19 28 38 50 63 76 91 106 123 141 16015 4 11 20 29 40 52 65 79 94 110 127 145 164 18516 4 12 21 31 42 54 67 82 97 114 131 150 169 19017 5 12 21 32 43 56 70 84 100 117 135 154 175 19518 5 13 22 33 45 58 72 87 103 121 139 159 179 20119 5 13 23 34 46 60 74 90 107 124 144 163 184 20520 5 14 24 35 48 62 77 93 110 128 148 168 189 21121 6 14 25 37 50 64 79 95 114 132 152 172 194 21622 6 15 26 38 51 66 82 99 117 136 156 177 199 22223 6 15 27 39 53 68 85 102 120 139 160 181 203 22624 6 16 28 40 55 70 87 104 123 143 164 185 208 23225 6 16 28 42 57 72 89 107 126 146 168 190 213 23726 7 17 29 43 58 74 92 110 129 150 172 194 218 24227 7 17 31 45 60 76 94 113 133 154 176 199 223 24728 7 19 32 46 62 78 96 116 136 157 180 203 228 253
α = 1%
n1 2 3 4 5 6 7 8 9 10 11 12 13 14 15n25 156 10 16 237 10 17 24 318 11 17 25 34 439 6 11 18 26 35 45 5610 6 12 19 27 37 47 58 7111 6 12 20 28 38 49 61 74 8712 7 13 21 30 40 51 63 76 90 10613 7 14 22 31 41 53 65 79 93 109 12514 7 14 22 32 43 54 67 81 96 112 129 14715 8 15 23 33 44 56 70 84 99 115 133 151 17116 8 15 24 34 46 58 72 86 102 119 137 155 17517 8 16 25 36 47 60 74 89 105 122 140 159 17918 8 16 26 37 49 62 76 92 108 125 144 163 18419 3 9 17 27 38 50 64 78 94 111 128 147 167 18820 3 9 18 28 39 52 66 81 97 113 132 151 171 19321 3 9 18 29 40 53 68 83 99 116 135 155 175 19722 3 10 19 29 42 55 70 85 102 119 138 158 179 20123 3 10 19 30 43 57 71 87 104 122 142 162 184 20624 3 10 20 31 44 58 73 89 107 125 145 166 188 21025 3 11 20 32 45 59 75 91 109 128 148 170 192 21526 3 11 21 32 46 60 76 94 112 131 152 173 196 22027 4 11 21 33 47 62 78 96 115 134 155 177 200 22428 4 11 21 34 48 63 80 98 117 137 159 181 204 229
30
4.2 Test unilateral
On rejette H0 : FX = FY par rapport a :– H1 : FX > FY si WX ≤ B ;– H1 : FX < FY si WX ≥ n1(n1 + n2 + 1) − B,
B etant la valeur lue dans la table.
α = 5%
n1 2 3 4 5 6 7 8 9 10 11 12 13 14 15n223 3 64 3 6 115 3 7 12 196 3 8 13 20 287 3 8 14 21 29 398 4 9 15 23 31 41 519 4 9 16 24 33 43 54 6610 4 10 17 26 35 45 56 69 8211 4 11 18 27 37 47 59 72 86 10012 5 11 19 28 38 49 62 75 89 104 12013 5 12 20 30 40 52 64 78 92 108 125 14214 5 13 21 31 42 54 67 81 96 112 129 147 16615 6 13 22 33 44 56 69 84 99 116 133 152 171 19216 6 14 24 34 46 58 72 87 103 120 138 156 176 19817 6 15 25 35 47 61 75 90 106 123 142 161 183 20318 7 15 26 37 49 63 77 93 110 127 146 167 188 21019 7 16 27 38 51 65 80 96 113 131 151 171 193 21520 7 17 28 40 53 67 83 99 117 136 156 176 198 22121 9 19 30 42 56 71 86 103 121 140 160 181 203 22622 9 19 31 44 58 73 89 106 125 144 164 186 208 23223 10 20 32 45 59 75 92 109 128 147 169 190 213 23724 10 21 33 47 61 77 94 112 131 152 173 195 219 24325 10 21 34 48 63 79 97 115 135 155 177 200 224 24826 11 22 35 49 65 82 100 118 138 160 182 205 229 25427 11 23 36 50 67 83 102 121 142 163 186 209 234 25928 11 23 37 52 69 86 105 125 145 167 190 214 239 265
α = 1%
n1 2 3 4 5 6 7 8 9 10 11 12 13 14 15n2234 95 10 166 5 11 17 247 6 11 18 25 348 6 12 19 27 35 459 7 13 20 28 37 47 5910 7 13 21 29 39 49 61 7411 7 14 22 30 40 51 63 77 9112 2 8 15 23 32 42 53 66 79 94 10913 3 8 15 24 33 44 56 68 82 97 113 13014 3 8 16 25 34 45 58 71 85 100 116 134 15215 3 9 17 26 36 47 60 73 88 103 120 138 156 17616 3 9 17 27 37 49 62 76 91 107 124 142 161 18117 3 10 18 28 39 51 64 78 93 110 127 146 165 18518 3 10 19 29 40 52 66 81 96 113 131 150 170 19119 4 10 19 30 41 54 68 83 99 116 135 153 174 19520 4 11 20 31 43 56 70 85 102 120 138 158 179 20021 4 11 21 32 44 58 72 88 105 122 142 161 183 20522 4 11 21 33 45 59 74 91 108 126 145 166 187 21023 4 12 22 34 47 61 77 93 111 129 149 169 192 21424 4 12 23 35 48 63 79 95 113 133 153 174 196 21925 4 13 23 36 49 64 81 97 116 135 156 177 201 22426 4 13 24 37 51 66 83 100 119 139 160 182 205 22927 5 13 25 37 52 67 85 102 122 142 164 185 209 23328 5 13 25 39 54 70 87 105 125 145 l67 190 214 239
31
5 Test de Wilcoxon signe
Soit M la mediane de Y − X , et W+ la somme des rangs des differences positives. On rejetteH0 : M = 0 par rapport a :
– H1 : M < 0 si W+ ≤ B ;– H1 : M > 0 si W+ ≥ n(n + 1)/2 − B ;– H1 : M = 0 si W+ ≤ B ou W+ ≥ n(n + 1)/2 − B ,
B etant la valeur lue dans l’une des tables ci-dessous (test unilateral ou bilateral).
Test bilateral Tests unilaterauxn risque 5% risque 1% n risque 5% risque 1%6 0 6 27 2 7 28 3 0 8 59 5 1 9 8 210 8 3 10 10 411 10 5 11 13 712 13 9 12 17 913 17 9 13 21 1214 21 12 14 25 1515 25 15 15 30 1916 29 19 16 35 2317 34 23 17 41 2718 40 27 18 47 3219 46 32 19 53 3720 52 37 20 60 4321 59 43 21 68 4822 66 49 22 75 5323 73 55 23 83 6124 81 61 24 92 6825 89 68 25 101 76
32
6 Distribution de Kolmogorov-Smirnov
dn,1−α
1 − α .80 .85 .90 .95 .99n1 .900 .925 .950 .975 .9952 .684 .726 .776 .842 .9293 .565 .597 .642 .708 .8294 .494 .525 .564 .624 .7345 .446 .474 .510 .563 .669
6 .410 .436 .470 .521 .6187 .381 .405 .438 .486 .5778 .358 .381 .411 .457 .5439 .339 .360 .388 .432 .51410 .322 .342 .368 .409 .486
11 .307 .326 .352 .391 .46812 .295 .313 .338 .375 .45013 .254 .302 .325 .361 .43314 .274 .292 .314 .349 .41815 .266 .283 .304 .338 .404
16 .258 .274 .295 .328 .39117 .250 .266 .286 .318 .38018 .244 .259 .278 .309 .37019 .237 .252 .272 .301 .36120 .231 .246 .264 .294 .352
25 .21 .22 .24 .264 .3230 .19 .20 .22 .242 .2935 .18 .19 .21 .23 .2740 .21 .2550 .19 .23
60 .17 .2170 .16 .1980 .15 .1890 .14100 .14∞ 1.07√
n1.14√
n1.22√
n1.36√
n1.63√
n
33
7 Formulaire
Probabilites
DefinitionsExperience aleatoire experience dont le resultat ne peut etre prevu a prioriEspace fondamental ensemble des resultats d’une experience aleatoire (souvent note Ω)
Evenement aleatoire evenement vrai ou faux suivant le resultat d’une experience aleatoire (⊂ Ω)
Tribu A sur Ω Ω ∈ A A ∈ A ⇒ A ∈ A⋃
n∈NAn ∈ A
Probabilite sur (Ω, A) P : A → [0, 1] tq P(Ω) = 1 et Ai incompatibles ⇒ P(⋃
Ai) =∑
P(Ai)
Proba. conditionnelle P(A|B) = P(A∩B)P(B)
Independance A et B ind. si P(A ∩ B) = P(A)P(B)Indep. mutuelle A1, . . . , An mut. ind. si ∀I ⊂ 1, . . . , n =⇒ P(
⋂i∈I Ai) =
∏i∈I P(Ai)
Proprietes
P(∅) = 0 P(A) = 1 − P(A) A ⊂ B ⇒ P(A) ≤ P(B)P(A ∪ B) = P(A) + P(B) − P(A ∩ B) P(∪Ai) ≤
∑P(Ai)
Th. de Bayes P(B|A) =P(A|B)P(B)
P(A) et (B1, . . . , Bn) partition de Ω ⇒ P(Bi|A) =P(A|Bi)P(Bi)∑j P(A|Bj )P(Bj )
Variables aleatoiresVariable aleatoire application mesurable de (Ω, A, P ) dans (R,B)
Loi de probabilite PX(B) = P(ω ∈ Ω|X(ω) ∈ B) = P(X−1(B)) notee P(X ∈ B)discret : p(x) = P(X = x) et P(X ∈ B) =
∑x∈B p(x)
continu : densite f et P(X ∈ I) =∫
If(x)dx
F. de repartition F (x) = P(X ≤ x), F continue a droite et croissante de de 0 a 1, F ′ = f pour 1 v.a. continueF. d’1 v.a. ϕ(X) discret : p(a) =
∑x|ϕ(x)=a p(x)
continu : G = F ϕ−1 (ϕ strictement crois.) ou G = 1 − F ϕ−1 (ϕ strictement dec.)Esperance E(X) =
∑xp(x) ou
∫xf(x)dx et E(ϕ(X)) =
∑ϕ(x)p(x) ou
∫ϕ(x)f(x)dx
Variance et covariance Var(X) = E([X − E(X)]2) = E(X2) − [E(X)]2
Cov(X, Y ) = E[(X − E(X))(Y − E(Y ))] = E(XY ) − E(X)E(Y )
Moments d’ordre k non centre mk = E(Xk) , centre µk = E([X − E(X)]k)V. a. independantes discret : p(x1, . . . , xn) = p(x1) . . . p(xn)
continu : f(x1, . . . , xn) = f(x1) . . . f(xp) ou F (x1, . . . , xn) =∏n
i=1 F (xi)E(X1 . . . Xn) = E(X1) . . . E(Xn), Cov(X, Y ) = 0, Var(
∑Xi) =
∑Var(Xi)
Lois de probabilites
Lois discretesLoi notations p(x) Domaine E(X) Var(X)
uniforme U(n) 1/n 1, . . . , n (n + 1)/2 (n2 − 1)/12Bernoulli B(1, p) px(1 − p)1−x 0, 1 p p(1 − p)
binomiale B(n, p) Cxnpx(1 − p)n−x 0, . . . , n np np(1 − p)
Poisson P(λ) e−λ λx
x! N λ λ
Lois continuesLoi notations f(x) Domaine E(X) Var(X)
uniforme U[a,b]1
b−a 1[a,b](x) Ra+b2
(b−a)2
12
normale N (µ, σ2) 1√2πσ2
e− 12 ( x−µ
σ)2
R µ σ2
chi-deux χ2n R+ n 2n
∑ n1 (N (0, 1))2
exponent. E(θ) θe−θxR+ 1/θ 1/θ2
Student Tn R 0 (n > 1) nn−2 (n > 2) N (0, 1)/
√χ2
nn
Fisher Fn,m R+m
m−22m2(n+m−2)
n(m−4)(m−2)2(
χ2n
n )/(χ2
mm )
Convergence stochastique
Definitions
en probabilite (Xn)P→ a ∀ε et η, ∃n0 tel que n > n0 entraıne P(|Xn − a| > ε) < η
(Xn)P→ X (Xn − X)
P→ 0
en loi (Xn)L→ X Fn(x) → F (x) en tout point x de continuite de F
ProprietesCvg en probabilite =⇒ Cvg en loi
E(Xn) → a et Var(Xn) → 0 =⇒ (Xn)P→ a
Th. de Slutsky :Xn
L−→ X
YnP−→ a
=⇒
⎧⎪⎪⎪⎨⎪⎪⎪⎩Xn + Yn
L−→ X + a
XnYnL−→ aX
Xn
Yn
L−→X
asi a = 0.
Theoreme de la limite centrale
Soit (Xn) une suite de v.a. iid, d’esperance µ et de variance σ2. On a : Xn−µσ/
√n
L−→ N (0, 1) avec Xn = 1n
∑ ni=1 Xi.
34
Echantillon
Statistiques usuelles d’un echantillon X1, . . . , Xn
X = 1n
∑i Xi S∗2 = 1
n−1
∑i(Xi − X)2 = 1
n−1 (∑
i X2i − nX
2)
F (x) = 1n cardi : Xi ≤ x
Fractile empirique : fα =
X(nα) si nα ∈ N,X(nα+1) sinon.
Fonctions pivotales associees a un echantillon gaussien de taille n
µ X−µσ√n
∼ N (0, 1) si σ2 connue X−µS∗√
n
∼ Tn−1 si σ2 inconnue
σ2∑
(Xi−µ)2
σ2 ∼ χ2n si µ connue (n−1)S∗2
σ2 ∼ χ2n−1 si µ inconnue
Fonctions pivotales associees a 2 echantillons gaussiens independants de taille n et m
(S∗2
Xσ2
X
)/(S∗2
Yσ2
Y
) ∼ Fn−1,m−1X−Y −(µX−µY )
S∗√1n
+ 1m
∼ TN−2 (si meme variance)
Estimation
Precision d’un estimateur E[(θ − θ)2]
Borne de Frechet (u′(θ))2
In(θ) ou In(θ) = E[( ∂ ln L
∂θ )2]
= −E( ∂2 ln L∂θ2 )
CNS d’efficacite : ∂ ln L∂θ (θ; X1, . . . , Xn) = A(n, θ)(u − u(θ)) (on a Var(u) = u′(θ)
A(n,θ) )
Tests
Tests non parametriques
E(WX) = n(n+m+1)2 et Var(WX ) = nm(n+m+1)
12E(W+) = n(n+1)
4 et Var(W+) = n(n+1)(2n+1)24
Test du χ2
D2 =∑ K
k=1(Nk−npk0)
2
npk0=
∑ Kk=1
N2k
npk0− n
H0∼ χ2K−1
Tableaux de contingence : D2 =∑ r
i=1∑ s
j=1
(Nij− Ni.N.j
n
)2
Ni.N.jn
=∑ r
i=1∑s
j=1
N2ij
Ni.N.jn
− nH0∼ χ2
(r−1)(s−1)
Test de Kolmogorov-Smirnov
Dn = max1≤i≤n max(∣∣∣F (xi) − F0(xi)
∣∣∣ ,∣∣∣F (x−
i ) − F0(xi)∣∣∣)
Test de normalite
Region critique pour α = 0.05 : (√
n + 0.85√n
− 0.01)Dn > 0.895
Region critique pour α = 0.01 : (√
n + 0.85√n
− 0.01)Dn > 1.035
Analyse de la variance
Region critique du test de Bartlett : (N − K) ln(MSW ) −∑ K
k=1(nk − 1) ln(S∗2k ) > χ2
K−1,1−α
SSW =∑
k
∑i(X
ik − Xk)2 et MSW = SSW
N−K
SSB =∑
k nk(Xk − X)2 et MSB = SSBK−1
Sous H0 : MSBMSW ∼ FK−1,N−K
Procedure LSD : µk et µl significativement differents si|Xk−Xl|√
MSW (1/nk+1/nl)> tN−K;1−(α∗/2)
Regression
b =SxYS2
xet a = Y − SxY
S2x
x
a ∼ N (a, σ2n (1 + x2
S2x
)) et b ∼ N (b, σ2
nS2x
)
S2Y = Sreg + Sres avec Sreg = 1
n
∑ ni=1(Yi − Y )2 = b2S2
x et Sres = 1n
∑ ni=1(Yi − Yi)
2 = 1n
∑ ni=1 ε2
i
σ2MV = Sres et σ2 = n
n−2Sres
Intervalle de confiance sur E(Y0) : Y0 ± tn−2;1− α2
σ
√1n +
(x0−x)2
nS2x
Intervalle de prediction : Y0 ± tn−2;1− α2
σ
√1 + 1
n +(x0−x)2
nS2x
35
